Section outline


  • Key unit competence: Analyze electric field potential and gravitational 
    potential.
     
    Unit Objectives:
     By the end of this unit, I will be able to;
     ◊   list the properties of an electric and gravitational fields and the 
    variation of potentials properly.
     ◊   explain the working mechanism of a cathode ray tube, TV tubes 
    and computer monitors properly.
     ◊   explain the everyday applications of electric and magnetic fields.

    7.0 INTRODUCTION 
    Electricity might be leading technological advancement, but its study began 
    with nature. Electrical storms are a very dramatic example of natural 
    phenomena involving electricity. Other examples are found in animals. 
    Some use electricity as a tool for survival – as a weapon (by electric eels) or 
    to sense live food (by platypus and sharks). Animals routinely use electricity 
    to control their bodies. The story of Frankenstein’s monster, brought to 
    life during an electrical storm, was inspired by early experiments where 
    the legs of a dead frog were made to twitch by sending electrical current 
    through them. Today we use electrical technology not just to support our 
    everyday lives in a myriad of ways, but also to diagnose muscle and nerve 
    activity inside the body, and to assist faulty signaling in the body.
     
    7.1 ELECTRIC POTENTIAL
     7.1.1 Electric field and Coulomb’s law
     When a small charged particle is located in the area surrounding a 
    charged object, the charged particle experiences a force in accordance with 
    Coulomb’s Law. The space around the charged object where force is exerted 
    on the charged particle is called an electric field or electrostatic field. 
    Theoretically, an electric field due to charge extends to infinity but its effect 
    practically dies away very quickly as the distance from the charge increases.

         Electric field is a vector quantity whose direction is defined as the direction 
    which a positive test charge would be pushed when placed in the field. Thus, 
    the electric field direction about a positive charge is always directed away 
    from the positive source. And the electric field direction about a negative 

    charge is always directed toward the negative source as shown in Fig.7.1 


    Electric field exists at a point if a test charge at that point experiences an electric - force.


    The magnitude of the field is proportional to the number of field-lines per 
    unit area passing through a small surface normal to the lines. 

    The electric field strength or The electric field E at a point in space is defined 
    as the electric force Fe acting on a positive test charge q0 placed at that point 
    divided by the magnitude of the test charge:
     
    The electric field strength or The electric field E at a point in space is defined 
    as the electric force Fe acting on a positive test charge q0 placed at that point 

    divided by the magnitude of the test charge:


    We require the test charge to be small enough to have a negligible effect on 
    the charges on the sphere. A large test charge will cause a rearrangement of 
    the charges of the sphere due to induction and thus the test charge does not 

    have negligible effect on the sphere. 

    According to Coulomb’s law, the force exerted by q on the test charge is



    Thus gravitational filed  can be regarded as the gravitational force per 
    unit mass or the acceleration due to gravity. The gravitational field or 
    gravitational force per unit mass, is a useful concept because it does not 
    depend on the mass of the body on which the gravitational force is exerted; 
    likewise, the electric field or electric force per unit charge, is useful because 
    it does not depend on the charge of the body on which the electric force is 
    exerted
     
    Example 7.1 Electric field due a single point charge

     1. Calculate the magnitude and direction of the electric field at a point P 


    The direction of the electric field is toward the charge q as shown in Fig.7.3a, 
    since we defined the direction as that of the force on a positive test charge 
    which here would be attractive. 


    If q had been positive, the electric field would have pointed away, as in Fig. 7.3b.


    NOTE There is no electric charge at point P. But there is an electric field 
    there. The only real charge is q
     
    7.1.2 Electric potential and electric potential energy

     Electric potential is the potential energy per charge.


     The change in potential energy between any two points, a and b, equals the 
    negative of the work done by the conservative force on an object as it moves 
    from point a to point b.
     
    If we solve (7.01) and (7.03) for E, we find the general expression for potential 

    difference at a point located a distance d from the charge


    Example 7.2: Motion of a Proton in a Uniform Electric Field is only 

    valid for the case of a uniform electric field

     1. A proton is released from rest in a uniform electric field that has a 
    magnitude of  8.0 104 / V m and is directed along the positive x axis (Fig. 

    7.1). The proton undergoes a displacement of 0.50 m in the direction of E.


     (a) Find the change in electric potential between points A and B. 
    (b) Find the change in potential energy of the proton for this displacement.
     (c) Use the concept of conservation of energy to find the speed of the proton 
    at point B (after completing the 0.50 m displacement in the electric field) 
    (d) What if the situation is exactly the same as that shown in Figure, but 
    no proton is present? Could both parts (A) and (B) of this example still 

    be answered?


    The negative sign means the potential energy of the proton decreases as it
    moves in the direction of the electric field; it gains kinetic energy and at the 
    same time loses electric potential energy.
     
    (c) The charge–field system is isolated, so the mechanical energy of the 

    system is conserved:


     (d) Part (A) of the example would remain exactly the same because the 
    potential difference between points A and B is established by the source 
    charges in the parallel plates. The potential difference does not depend on 
    the presence of the proton, which plays the role of a test charge. 

    Part (B) of the example would be meaningless if the proton is not present. A 
    change in potential energy is related to a change in the charge–field system. 
    In the absence of the proton, the system of the electric field alone does not change

      Positive charge moving in opposite direction of electric field
     Now let us calculate the potential difference between two points A and B in 

    the field of a single positive charge q, see the Fig.7.5.


    When a unit test charge 
    is placed in electric field E created by some source 
    charge distribution at a distance   from the charge q placed at 0 in free 

    space the electric force acting on the test charge is given by.


    This force is conservative because the force between charges described by 
    Coulomb’s law is conservative. When the test charge is moved in the field by 
    some external agent, the work done by the field on the charge is equal to the 

    negative of the work done by the external agent causing the displacement. 

    The force is not constant during the displacement, the work done in taking 

    the charge from B to A, against the electric field E over short distance dr is 


    Thus the electric potential at an arbitrary point in an electric field 
    equals the work required per unit charge to bring a positive test charge 
    from infinity to that point 

    The potential near a positive charge is large and positive, and it decreases 
    toward zero at very large distances, Fig.7.6a. The potential near a negative 

    charge is negative and increases toward zero at large distances, Fig.7.6b. 


    Example 7.3: Work required to bring two positive charges close



    7.1.3 Equipotential Lines and Surfaces
     The electric potential can be represented by drawing equipotential lines 
    or equipotential surfaces. An equipotential surface is the one on which 
    all points are at the same potential. The potential difference between any 
    two points on the surface is zero, so no work is required to move a charge 
    from one point on the surface to the other. An equipotential surface 

    must be perpendicular to the electric field at any point. If this was not 


    The fact that the electric field lines and equipotential surfaces are mutually 
    perpendicular, helps us locate the equipotentials when the electric field lines 
    are known. In a normal two-dimensional drawing, we show equipotential 
    lines, which are the intersections of equipotential surfaces with the plane 

    of the electric field line.


    In Fig. 7.7, a few of the equipotential lines are drawn (dashed green lines) 
    for the electric field (red lines) between two parallel plates maintained at a 
    potential difference of 20 V. The negative plate is arbitrarily chosen to be 

    zero volts and the potential of each equipotential line is indicated.


    7.1.4 Potential due to electric dipole
     The field lines between two opposite and equal charges make what is called 

    a dipole. An electric dipole is a pair of point charges with equal magnitude 


    The equipotential lines for the case of two equal but oppositely charged 
    particles are shown in Fig. 7.8 as green dashed lines.
     
    Unlike electric field lines, which start and end on electric charges, 
    equipotential lines and surfaces are always continuous curves, and continue 

    beyond the borders indicated in Figs. 7.7 and 7.8.


     Electric Potential Energy with Several Point Charges 

    We obtain the electric potential resulting from two or more point charges by 
    applying the superposition principle. That is, the total electric potential 
    at some point P due to several point charges is the sum of the potentials 
    due to the individual charges. For a group of point charges, we can write the 

    total electric potential at P in the form: 


     If the system consists of more than two charged particles, we can obtain 
    the total potential energy by calculating U for every pair of charges and 
    summing the terms algebraically. As an example, the total potential energy 

    of the system of three charges shown in Fig.7. 10 is








    7.1.5 Conservation of electrical energy
     Energy is conserved in the movement of a charged particle through an 
    electric field, as it is in every other physical situation. Electric charge 
    cannot be created or destroyed (though positive and negative charges can 
    neutralise each other).

    Given a stationary test charge at a certain location, an applied electric field 
    will cause the charge to move to one end or the other, depending on the charge.
     Positive test charges will move in the direction of the field; negative charges 
    will move in the opposite direction.

     At the instant at which the field is applied, the motionless test charge has 
    zero kinetic energy, and its electric potential energy is at the maximum. 
    Now the charge accelerates, and its kinetic energy (due to motion) increases 
    as its potential energy decreases. The sum of energies is always constant.

    The formula illustrating conservation of energy can be written in many 

    ways, but all expressions are based on the simple premise of equating the 



    Application Activity 7.1



    7.2 ELECTRODYNAMICS
     This is the study of phenomena associated with charged bodies in motion 
    and varying electric and magnetic fields. Since a moving charge produces a 
    magnetic field, electrodynamics is concerned with effects such as magnetism, 
    electromagnetic radiation and electromagnetic induction, including some 
    practical applications as the electric generator and the electric motor. 

    This area of electrodynamics, often known as classical electrodynamics, 
    was first systematically explained by the physicist James Clarke Maxwell. 
    Maxwell’s equations, a set of differential equations, describe the phenomena 
    of this area with great generality. A more recent development is quantum 
    electrodynamics, which was formulated to explain the interaction of 
    electromagnetic radiation with matter, to which the laws of the quantum 
    theory apply. 

    When the velocities of the charged particles under consideration become 
    comparable with the speed of light, corrections involving the theory of 
    relativity must be made; this branch of the theory is called relativistic 
    electrodynamics. It is applied to phenomena involved with particle 
    accelerators and with electron tubes that are subject to high voltages and 
    carry heavy currents.
     
    7.2.1 Cathode ray tube
     The CRT is a vacuum tube in which a beam of electrons is accelerated and 
    deflected under the influence of electric or magnetic fields. The electron 
    beam is produced by an assembly called an electron gun located in the 
    neck of the tube. These electrons, if left undisturbed, travel in a straight
    line path until they strike the front of the CRT, the “screen’’, which is coated 
    with a material that emits visible light when bombarded with electrons.

    The operation of a CRT depends on thermionic emission, discovered 
    by Thomas Edison (1847–1931). Consider a voltage applied to two small 
    electrodes inside an evacuated glass “tube” as shown in Fig. 7.7: the cathode 
    is negative, and the anode is positive. If the cathode is heated (usually by 
    an electric current) so that it becomes hot and glowing, it is found that 
    negative charges leave the cathode and flow to the positive anode. These 
    negative charges are now called electrons, but originally they were called 
    cathode rays because they seemed to come from the cathode.

    Fig.7.13 is a simplified sketch of a CRT which is contained in an evacuated 
    glass tube. A beam of electrons, emitted by the heated cathode, is accelerated 
    by the high-voltage anode and passes through a small hole in that anode. The 

    inside of the tube face on the right (the screen) is coated with a fluorescent 

    material that glows at the spot where the electron hits. Voltage applied 
    across the horizontal and vertical deflection plates can be varied to deflect 
    the electron beam to different spots on the screen. The instruments used in 
    the laboratory to display, measure and analyse the waveforms of different 

    circuits is known as cathode ray oscilloscope.


    7.2.2 TV and computer monitors
     In TV and computer monitors, the CRT electron beam sweeps over the 
    screen in the manner shown in Fig.7.14 by carefully synchronizing voltages 
    applied to the deflection plates. This is called scanning.
     During each horizontal sweep of the electron beam, the grid receives a 
    signal voltage that limits the flow of electrons at each instant during the 
    sweep; the more negative the grid voltage is, the more electrons are repelled 
    and fewer pass through, producing a less bright spot on the screen. Thus, 
    the varying grid voltage is responsible for the brightness of each spot on 
    the screen. At the end of each horizontal sweep of the electron beam, the 
    horizontal deflection voltage changes dramatically to bring the beam back 
    to the opposite side of the screen, and the vertical voltage changes slightly 
    so the beam begins a new horizontal sweep slightly below the previous one. 
    The difference in brightness of the spots on the screen forms the “picture”. 

    Colour screens     have red, green, and blue phosphors which glow when 
    struck by the electron beam. The various brightnesses of adjacent red, 
    green and blue phosphors (so close together we don’t distinguish them) 
    produce almost any colour. With 30 new frames or pictures every second 
    (25 in countries with 50-Hz line voltage), a “moving picture” is displayed on 
    the TV screen. The commercial movies present 24 frames per second as the 

    film runs.


     7.2.3  Trajectory of a charge moving in a cathode ray tube
     If electrons enter an electric field in a CRT acting at right angles to their 
    direction of motion, they are deflected from their original path. In Fig. 7.15, 
    a p.d is applied between the plates P and Q of length l, creates an electric 
    field of intensity E. Consider an electron of charge e, mass m and velocity v 

    entering the field.


    The value of z is measured from the centre of plates. Assume that the 
    separation of plates is d.

     Field intensity E is given by;


     Since E is vertical, there is no horizontal force acting on the electron. Hence, 

    the horizontal velocity is not affected, i.e. it remains constant.






    Application Activity 7.2
    1. Fig. 7.17 shows two metal plates 2.0 cm long placed 5 mm apart. 
    A fluorescent screen is placed 20.0 cm from one of the plates. An 
    electron of kinetic energy 3.2 × 10–6 J is incident mid-way between 
    the plates. Calculate the voltage applied across the plates to deflect 
    the electron 2.1 cm on the screen. Assume that the electron moves 

    through vacuum.


     2. In the diagram of Fig. 7.18, P and Q are parallel metal plates each 
    of length l = 4 cm. A p.d of 12V is applied between P and Q. The 
    space between P and Q is virtual. A beam of electrons of speed 1.0 
    × 106 m/s is directed mid-way between P and Q at right angles to 
    the electric field between P and Q. Show that the electron beam 
    emerges from the space between P and Q at an angle of 64.6° to the 

    initial direction of the beam.


    7.3 GRAVITATIONAL ENERGY
     7.3.1 Newton’s Law of Universal Gravitation
     In 1687 Newton published his work on the law of gravity in his treatise 
    Mathematical Principles of Natural Philosophy. Newton’s law of universal 
    gravitation states that

    Every particle in the Universe attracts every other particle with a force 
    that is directly proportional to the product of their masses and inversely 
    proportional to the square of the distance between them.
     
    If the particles have masses m1     and  m2 and are separated by a distance r, 
    (Fig.7.24) the magnitude of this gravitational force is

    The form of the force law given by Equation 7.43 is often referred to as an 
    inverse square law because the magnitude of the force varies as the inverse 

    square of the separation of the particles.1


    The magnitude of the force exerted by the Earth on a particle of mass m 

    near the Earth’s surface is


     This force is directed toward the center of the Earth.

    7.3.2 Gravitational potential energy

    Gravity is a conservative force, and we may define a potential energy 
    associate with it. Recall that the work you must do to lift a mass m from 
    one point to another is equal to the gain in potential energy. Work is done 

    against gravity only when the displacement is radial. Going sideways to r 


     The work done by the gravitational force when the body moves directly 

    away from or toward the center of the earth is given by:


    We define the corresponding gravitational potential energy U so that 


    This approximation is useful near the surface of the earth.
    If the potential at infinity is taken as zero by convention, the negative sign 

    indicates that the potential at infinity (zero) is higher than the potential 


    The gravitational potential energy of a body of mass m due to the Earth’s 
    gravitational field is zero at infinity; when a body moves from infinity to a 
    point in the gravitational field, its potential energy decreases and kinetic 
    energy increases as shown in Fig.7.32b. Although Equation 7.42 was derived 

    for the particle–Earth system, it can be applied to any two particles. That 

    is, the gravitational potential energy associated with any pair of particles 
    of masses m1 and m2 separated by a distance r is

    When two particles are at rest and separated by a distance r, an external
    agent has to supply energy at least equal to in order to 
    separate the particles to an infinite distance. 

    It is therefore convenient to think of the absolute value of the potential 
    energy as the binding energy of the system. If the external agent supplies 
    energy greater than the binding energy, the excess energy of the system 
    will be in the form of kinetic energy when the particles are at an infinite 

    separation. 

    Example 7.7: Binding energy
     1. Calculate the binding energy of the earth-sun system neglecting the 
    effect of the presence of other planets and satellites. Mass of earth = 6 × 10 24,
    mass of sun = 3.3 ×10     5 times the mass of earth and the distance between 

    earth and sun = 1.5 × 10 8 km. 

    Answer: 

    The binding energy is the absolute value of the potential energy


    We can extend this concept to three or more particles. In this case, the total
    potential energy  of the system is the sum over all pairs of particles. 
    Each pair contributes a term of the form given by Equation 7.36. For 
    example, if the system contains three particles, as in Fig.7.29, we find that 

    by superposition principle.


    The absolute value of  represents the work needed to separate the 
    particles by an infinite distance.
     
    Example 7.8: Superposition of gravitational potential energy
     1. A system consists of three particles, each of mass 5.00 g, located at 
    the corners of an equilateral triangle with sides of 30.0 cm. (a) Calculate 
    the potential energy of the system. (b) If the particles are released 

    simultaneously, where will they collide?

    Answer


    The gravitational potential energy of the system is the sum of the gravitational 

    potential energies of all three pairs of particles



    The total mechanical energy in a circular orbit is negative and equal to one
    half the potential energy. Increasing the orbit radius r means increasing 
    the mechanical energy (that is, making E less negative). Fig.7.35 shows 
    the variation of K, U, and E with r for a satellite moving in a circular orbit 
    about a massive central body. Note that as r is increased, the kinetic energy 

    (and thus also the orbital speed) decreases.


    If the satellite is in a relatively low orbit that encounters the outer fringes 
    of earth’s atmosphere, mechanical energy decreases due to negative work 
    done by the force of air resistance; as a result, the orbit radius decreases 

    until the satellite hits the ground or burns up in the atmosphere.

    Example 7.9
     1. A satellite of mass 450 kg orbits the Earth in a circular orbit at 6.83 Mm 
    above the Earth’s surface. Find: (a) the potential energy (b) the kinetic 
    energy and (c)  the total energy of the satellite
     
    Answer

     (a) the distance between the satellite and the center of the Earth is


    The total energy equals the negative of the kinetic energy.
     
    Escape speed: 
    Near the surface of the Earth, the force of attraction between the Earth 
    and some object is constant and equal to  which is independent of the 
    height of the object above the Earth’s surface. The gravitational field near 
    the surface of the Earth is said to be uniform. 

    If we project an object vertically upward with initial speed in uniform 
    gravitational field, it will rise to a maximum height given by the law of 

    conservation of mechanical energy:


    If we project an object upward with a very large initial speed so that the 
    object moves a distance comparable to the radius of the Earth, we must 
    take into account the decrease in the gravitational force on the object to 

    calculate correctly the maximum height the object attains.


    The minimum speed the object must have at the Earth’s surface in order to 
    escape from the influence of the Earth’s gravitational field is escape speed. 
    Traveling at this minimum speed, the object continues to move farther and 

    farther away from the Earth as its speed asymptotically approaches zero. 




     
    7.3.5 Relation between electric and gravitational field
     There are many similarities between Coulomb’s law and Newton’s law of 
    universal gravitation:
     • Both are inverse square laws that are also proportional to the product 
    of another quantity; for gravity it is the product of two masses, and for 
    the electric force it is the product of the two charges.
     • The forces act along the line joining the centres of the masses or charges.
     • The magnitude of the force is the same as the force that would be 
    measured if all the mass or charge is concentrated at a point at the 
    centre of the sphere.
     Therefore, distance in both cases is measured from the centres of the 
    spheres. In both cases we are assuming that r is longer than the radius of 
    the object. However, the two forces also differ in some important ways:
      The electric force can attract or repel, depending on the charges involved, 

    whereas the gravitational force can only attract.


    Just as a mass can be attracted gravitationally by more than one body at 

    once, so a charge can experience electric forces from more than one body 

    at once. Experiments have shown that the force between two charges can 
    be determined using Coulomb’s law independently of the other charges 
    present, and that the net force on a single charge is the vector sum of all 

    these independently calculated electric forces acting on it. 


    END OF UNIT ASSESSMENT
     1. Four particles of masses m, 2 m, 3 m and 4 m are kept in sequence at the 
    corners of a square of side a. Find the magnitude of gravitational force 

    acting on a particle of mass m placed at the centre of the square.


    2. Mass M is divided into two parts  xM and (1 – x)M. For a given separation, 
    the value of x for which the gravitational attraction between the two 
    pieces becomes maximum. Find this maximum value of x.

     3. Three identical point masses, each of mass 1 kg lies in the x – y plane at 
    points (0, 0), (0, 0.2 m) and (0.2 m, 0). Find the net gravitational force on 

    the mass at the origin.


     4. Two positive charges sit in an (x, y)-coordinate system. The first one has 
    charge q1
     = 0.40 µC and sits at (–0.30 m, 0). The second one has charge q2
     = 0.30 µC and sits at (0, +0.30 m). Find the electric potential at the origin.
     5. (a)  Find the electric potential energy of the system of two charges 

    shown in the Figure 7.26.


     (b) Find the electric potential energy of the system if a third charge 
    q3 = –0.10 µC is placed at the origin.
     
    6. Two rectangular copper plates are oriented horizontally with one directly 
    above the other. They are separated by a distance of 25 mm. The plates 
    are connected to the terminals a 5.0 volt flashlight battery. The positive 
    plate (the one at the higher electric potential) is at the bottom; the 

    negative plate (the one at the lower electric potential) is at the top.


    If an electron is placed on the upper plate, then released, with what speed 

    will it strike the lower plate? Use conservation of energy.

    7. A charge of +2.82 µC sits in a uniform electric field of 12.0 N/C directed 
    at an angle of 60° above the +x axis. The charge moves from the origin 

    (point A) to the point (1.40 m, 0) (point B) on the x-axis.


     •  a. Find the force exerted on the charge by the electric field.
     •  b. Find the work done on the charge by the electric field as the 
    charge moves from A to B.
     •  c. Find the change in the charge’s electric potential energy as it 
    moves from A and B.

     •  d. Find the electric potential difference between points A and B.

    UNIT SUMMARY

     Electric Field and Electric Potential Due to a Point Charge
     The direction of electric field is taken to be the direction of the force it would exert on 

    a positive test charge. 


    Electric Potential Energy and Potential Difference

    The work done by a conservative force in moving an object between any two 
    positions is independent of the path taken. Hence, we define the potential 

    energy for electrostatic force mathematically as:


    Equipotential Lines and Surfaces
     An equipotential surface is one on which all points are at the same potential. 
    An equipotential surface must be perpendicular to the electric field 
    at any point.
     
    Potential due to Electric Dipole
     Unlike electric field lines, which start and end on electric charges, 
    equipotential lines and surfaces are always continuous closed curved.
     
    Conservation of Electrical Energy
     At the instant at which the field is applied, the motionless test charge has 
    zero kinetic energy, and its electric potential energy is at a maximum. Then, 
    the charge accelerates, and its kinetic energy (from motion) increases as its 

    potential energy decreases. The sum of energies is always constant.


    Cathode Ray Tube (CRT)
     The CRT is a vacuum tube in which a beam of electrons is accelerated and 
    deflected under the influence of electric or magnetic fields. 
    These electrons, if left undisturbed, travel in a straight-line path until they 
    strike the screen of the CRT, which is coated with a material that emits 
    visible light when bombarded with electrons.
     
    TV and Computer Monitors
     In TV and computer monitors, the CRT electron beam sweeps over the 
    screen in the manner of carefully synchronized voltages applied to the 
    deflection plates and is called scanning.
     
    Trajectory of a charge moving in a cathode ray tube
     The equation of motion of a charge in a field is calculated by considering 

    vertical and horizontal displacements and is given by:


    This equation shows that when electron is in the field, its path is parabolic 
    and is called the equation of trajectory.
     
    The vertical deflection D of electron on the screen from initial direction of 

    motion can be obtained by using equation:


    Electrodynamics
     When the velocities of the charged particles under consideration become 
    comparable with the speed of light, corrections involving the theory of 
    relativity must be made; this branch of the theory is called relativistic 
    electrodynamics. 

    Gravitational Potential
     The gravitational potential V at a point is defined numerically as work done 

    in taking a uniform mass from infinity to that point. 


    Escape Velocity for a Planet
     If the rocket is fired from the surface of the earth with velocity v such that 
    it just escapes from the influence of the earth’s gravitational pull. Then this 

    velocity is called escape velocity.


    Energy Conservation in Gravitational Fields
     Conservation of energy tells us that the total energy of the system is 
    conserved, and in this case, the sum of kinetic and potential energy must 
    be constant. This means that every change in the kinetic energy of a system 
    must be accompanied by an equal but opposite change in the potential energy.