consider three point particles lying along the x axisseparatedby a distance d, as shown. These particles have charges-Q, +2Q, and-Q respectively. Note that the z axis is comingstraight up out ofthe paper from the location of the positivepoint charge.

(a) Calculate the electrostatic potential (y,.z) producedbythese charges at the arbitrary point [0,y,z] in the yzplane.

(b) Use your answer from part (a) along with asymmetryargument to calculate E (electric field) at the samearbitrarypoint. [ Note that if you like, you can check your answersfor E byusging the principle of superposition and the firld createdby apoint charge.

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Consider a ring of charge with total charge Q and with a radiusof R, centered at the origin in the x-y plane. Determine theelectric field at any location along the z-axis. Fill in theBLANKS.

1. Because of symmetry, the electric field at any point on thez-axis will always be in the _________________

2. Therefore, we only need to consider the ______________________of any contribution to the electric field from a single element ofcharge on the ring.

3. To calculate the electric field at a point on the z-axis, we aregoing to integrate around the ring of charge. The radius of thering is constant and therefore it makes sense for the variable ofintegration to be ___________________.

4. The z-component of the electric field from a single element ofcharge (dQ) is
_______________________________________ (write in terms of dQ, zand r-the distance from the element of charge to the pointp).

5. To calculate the charge on dQ we need to know the charge perunit length which is given by__________________________________________

6. An element of charge, dQ, has a length of _____________ (answerin terms of R and Î) and a charge of______________________________.

7. The distance from this element to the point p is given by:______________________

8. Rewriting # 4 in terms of #6 and #7 we get that_______________________________

9. Now to find the total electric field for a point along thez-axis we simply integrate around the ring making_____________________________________________

10. Suppose that z was really really large (much larger than R).What would you expect the electric field to behave as?

11. Suppose that z was zero. What do you expect the electric fieldto be?

12. Does your solution match these limiting situations?Explain.

1.Two uncharged, conducting spheres, A and B, are held at rest oninsulating stands and are in contact. A positively charged rod isbrought near sphere A. While the rod is in place, the two spheresare separated. How will the spheres be charged, if at all, afterthey are separated?
Sphere A Sphere B
(a) positive positive
(b) positive negative
(c) negative positive
(d) negative negative
(e) zero zero

2. A charged conductor is brought near an uncharged insulator.Which one of the following statements is true?
(a) Both objects will repel each other.
(b) Both objects will attract each other.
(c) Neither object exerts an electrical force on the other.
(d) The objects will repel each other only if the conductor has anegative charge.
(e) The objects will attract each other only if the conductor has apositive charge.

3. An electron is moved from point A, where the electric potentialis VA = â240 V, to point B, where the electric potential is VB =â360 V. What is the change in the electric
potential energy?
(a) 1.92Ã10â17 J (b) 2.04Ã10â17 J (c) 5.75Ã10â17 J (d) 8.98Ã10â17 J(e) 1.05Ã10â16 J

4. A -2q charge is located a distance d to the right of a +4qcharge, along the x
axis. At which point (or points) along the axis does the electricfield go to zero? To
solve this problem, consider an observer at point P, which is anarbitrary distance x from the +4q charge. Find the value (orvalues) for x (in terms of the separation distance d) where thetotal electric field vanishes.
(a) The electric field is never zero in the vicinity of thesecharges.
(b) The electric field is zero at x = -0.328d.
(c) The electric field is zero at x = +4.22d.
(d) The electric field is zero at both x = -2.26d and x =+2.26d.
(e) The electric field is zero at x = +3.41d.

5. point A is a distance L away from a point charge Q. Point B is adistance 4L away from Q. What is the ratio of the electrostaticpotential at B to that at A, i.e., what is VB/VA?
(a) 1/16
(b) 1/9
(c) 1/4
(d) 1/3
(e) can not be determined without specifying Q and L

6. Which one of the following statements best explains why it ispossible to define an electric potential
in a region of space that contains an electric field?
(a) Work must be done to bring two positive charges closertogether.
(b) Like charges repel one another and unlike charges attract oneanother.
(c) A positive charge will gain kinetic energy as it approaches anegative charge.
(d) The work required to bring two charges together is independentof the path taken.
(e) A negative charge will gain kinetic energy as it moves awayfrom another negative charge.

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