# Homework Solutions

## Problems: X1, X2, X3, X3, 17, 18, 21, 29, 31, 33, 36, 45

### from the FIFTH edition of Serway & Beichner

Questions:
Q1:
Distinguish between electric potential and electrical potential energy.

"Electric potential energy" is the energy of a particular charge q because of the other charges or fields that influence it. "Electric potential" refers to a regions of space and its characteristics because of charges or fields that influence it. "Electric potential" is the ratio of the electric potential energy that a charge q has at some point in space divided by that charge q.

Q3: Give a physical explanation of the fact that the potential energ of a pair of like charges is positive whereas the potential energy of a pair of unlike charges is negative.

Think back to the case of gravitational potential energy ( U = m g h ). Suppose that you choose or define the "reference point" for zero gravitational potential energy ( U = 0 ) to be on top of a table. If you have a book above this point it has positive gravitational potential energy and will fall back to U = 0 if you release it. If you have a book below this point it has negative gravitational potential energy and you must do work on it to get it back to U = 0.

Now, look at the electrical equivalent. We have choosen or defined U = 0 to be the state where the two particles are far, far apart. If we have two like charges that are near each other, they will go to this state if we release them. That is, they will repel each other and push each other far, far away. That means they had positive potential energy when they were nearby. That is not the case for two unlike charges. If we find them nearby, we must do work on them to get them to the state we have defined as U = 0. That means they had negative potential energy when they were nearby.

Q6: Describe the equipotential surfaces for
(a) an infinite line of charge and

The equipotential surfaces are cylinders, concentric with the line of charge.
(b) a uniformly charged sphere

The equipotential surfaces are spheres, concentric with the charged sphere.

Q8: The electric field inside a hollow, uniformly charged sphere is zero. Does this imply that the potential is zero inside the sphere?

No, the electric potential inside the charged sphere is constant. The electric field is the gradient or derivative of the electric potential. The derivative of a constant is zero.

Q14: Why is it important to avoid sharp edges or points on conductors used in high-voltage equipment?

The electric field near a conductor is inversely proportional to the radius of curvature of the surface. A sharp edge or point has a very small radius of curvature so the electric field near a sharp edge or point will be very large. This can cause to a breakdown of the air and sparking.

Problems:

25.X1 The gap between electrodes in a spart plug is 0.0060 cm. to produce an electric spark in a gasoline-air mixture, an electric field of 3.0 x 106 V/m must be achieved. When starting the car, what minimum voltage must be supplied by the ignition circuit?

25.X2 A deuteron (a nucleus that consists of one proton and one neutron) is accelerated through a 2.7 kV potential divverence.
(a) How much energy does it again?
(b) How fast is it going if it starts from rest?

25.X3 Consider two points in an electric field. The potential at P1 is V1 = - 30 V, and the potential at Pw is V2 = + 150 V. How much work is done by an external force in moving a chargee = - 4.7 microCoulombs from p2 to P1?

25.X4 The magnitude of the electric field between two charged parallel plates separated by 1.8 cm is 2.4 x 10 4 N/C. Find the potential difference between the two plates. How much kinetic energy is gained by a deuteron in accelerating from the positive to the negative plate?

25.X5 An electron in the beam of a typical television picture tube is accelerated through a potential difference of 20 kV before striking the face of the tube.
(a) What is the energy of this electron, in electron volts, and what is its speed when it strikes the screen?
(b) How much momentum is imparted to the screen by the electron?

25.X6 At what distance from a point charge of 8.0 microCoulombs does the electric potential equal 3.6 x 104 V?

25.29 A small spherical object carries a charge of 8.0 nanoCoulombs. At what distance from the center of the object is the potential equal to 100 V? 50 V? 25 V? Is the spacing of the equipotentials proportional
to the change in V?

25.18 A charge + 1 is at the origin. A charge - 2 q is at x = 2.0 m on the x-axis. For what finite vale(s) of x is
(a) the electric field zero?
(b) the electric ppotential zero?

25.31 In Rutherford’s famous scattering experiments that led to the planetary model of the atom, alpha particles (charge + 2 e, mass = 6.6 x 10 - 27 kg) were fired at a gold nucleus (charge + 79 e). An alpha particle initiallly very far from the gold nucleus is fired at 2.0 x 107
m/s directly toward the center of the nucleus. How close does the alpha particle get to this center before turning around?

25.33 Calculate the energy required to assemble the array of charges shown in Figure P25.32 where a = 0.20 m, b = 0.40 m, and q = 6.0 microCoulombs.

25.45 The potential is a region between x = 0 and x = 6.0 m is V = a + b x where a = 10 V and b = - 7.0 V/m. Determine
(a) the potential at x = 0, 3.0 m, and 6.0 m and
(b) the magnitude and direction of the electric field at x = 0, 3.0 m, and 6.0 m.

25.X7 Calculate the electric potential at point P on the axis of the annulus shown in Figure P25.49, which has a uniform charge density sigma.

Solution