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Chapter 2: Q59P (page 112)

Prove or disprove (with a counterexample) the following

Theorem:Suppose a conductor carrying a net charge Q,when placed in an

external electric field E鈫�e ,experiences a force F鈫�; if the external field is now

reversed ( localid="1657519836206" E鈫�e鈫�-E鈫�e), the force also reverses ( localid="1657519875486" F鈫�鈫�-F鈫�).

What if we stipulate that the external field isuniform?

Short Answer

Expert verified

If the external electric field near a conductor is reversed, the force on the conductor doesn't always get reversed.

Step by step solution

01

Step 1:Given data:

A conductor carrying a net charge Qis placed in an external electric field E鈫�e.

02

Charge induction on a conductor

A charge induces opposite charge on a conductor near it.

03

A spherical conductor placed in front of a point charge

Consider a conducting sphere placed in front of a positive point charge. The point charge induces negative charge on the closest surface of the sphere. The force on the sphere is thus towards the point charge.

If the point charge is negative, the field on the sphere changes sign. This time, positive charge is induced on the closest surface. The force is thus still towards the point charge.

Thus, in this case, the change in direction of the external electric field doesn't change the direction of force on the conductor. This disproves the theorem.

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Most popular questions from this chapter

Check that Eq. 2.29 satisfies Poisson's equation, by applying the Laplacian and using Eq. 1.102.

(a) Twelve equal charges, q,are situated at the comers of a regular 12-sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Qat the center?

(b) Suppose oneof the 12 q'sis removed (the one at "6 o'clock"). What is the force on Q?Explain your reasoning carefully.

(c) Now 13 equal charges, q,are placed at the comers of a regular 13-sided polygon. What is the force on a test charge Qat the center?

(d) If one of the 13 q'sis removed, what is the force on Q?Explain your reasoning.

Find the energy stored in a uniformly charged solid sphere of radiusRand charge q.Do it three different ways:

(a)Use Eq. 2.43. You found the potential in Prob. 2.21.

(b)Use Eq. 2.45. Don't forget to integrate over all space.

(c)Use Eq. 2.44. Take a spherical volume of radiusa.What happens as a鈫�鈭�?

Calculate the divergence of the following vector functions:

Two spheres, each of radius R and carrying uniform volume charge densities +p and -p , respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show that the field in the region of overlap is constant, and find its value. [Hint: Use the answer to Prob. 2.12.]

A conical surface (an empty ice-cream cone) carries a uniform surface charge .The height of the cone is as is the radius of the top. Find the potential difference between points (the vertex) and (the center of the top).
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