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Chapter 3: Q3.7P (page 129)

Find the force on the charge +qin Fig. 3.14. (The xyplane is a grounded conductor.)

Short Answer

Expert verified

Answer

The net force on q is -14蟺蔚0(29272d2)z.

Step by step solution

01

Given data

Consider the given figure as shown below.

Here, from the given figure it is clear that xyplane is grounded conductor, thus potential is zero

02

Determine force

As +qinduces an equal and opposite charge of -qat a distance of Z=-3dand -2qinduces +2qat distance of Z=-d.

Write the expression for force on +qdue to -2q,.

F1=14蟺蔚0q(-2q)(2d)2z 鈥︹�� (1)

Write the expression for force on +qdue to +2q,.

F1=14蟺蔚0q(-2q)(4d)2z 鈥︹�� (2)

Write the expression for force on +qdue to -q.

F1=14蟺蔚0q(-2q)(6d)2z 鈥︹�� (3)

03

Determine the net force

Write the expression for the net force on q .

F=F1+F2+F3=q4蟺蔚0[-2q2d2+2q4d2+-q6d2]z=q24蟺蔚0[-12+18-136]z=q24蟺蔚0(29q272d2)z

Thus, the net force on q is -q24蟺蔚0(29q272d2)z.

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Most popular questions from this chapter

Show that the average field inside a sphere of radius R, due to all the charge within the sphere, is

Eave=-14蟺蔚0蚁R3

Where 蚁is the total dipole moment. There are several ways to prove this delightfully simple result. Here's one method:

(a) Show that the average field due to a single chargeqat point r inside thesphere is the same as the field at r due to a uniformly charged sphere with

蚁=q/(43蟺R3), namely

14蟺蔚0(43蟺R3)鈭�qr2rd味'

Where r is the vector from r to d味

(b) The latter can be found from Gauss's law (see Prob. 2.12). Express the answerin terms of the dipole moment of q.

(c) Use the superposition principle to generalize to an arbitrary charge distribution.

(d) While you're at it, show that the average field over the volume of a sphere, dueto all the charges outside, is the same as the field they produce at the center.

A sphere of radiusR,centered at the origin, carries charge density

蚁(r,胃)=kRr2(R-2r)sin胃

where k is a constant, and r, 胃are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.

Solve Laplace's equation by separation of variables in cylindrical coordinates, assuming there is no dependence on z (cylindrical symmetry). [Make

sure you find all solutions to the radial equation; in particular, your result must accommodate the case of an infinite line charge, for which (of course) we already know the answer.]

A more elegant proof of the second uniqueness theorem uses Green's

identity (Prob. 1.61c), with T=U=V3. Supply the details.

In Ex. 3.2 we assumed that the conducting sphere was grounded ( V=0). But with the addition of a second image charge, the same basic modelwill handle the case of a sphere at any potentialV0 (relative, of course, to infinity). What charge should you use, and where should you put it? Find the force of attraction between a point charge q and a neutral conducting sphere.
See all solutions

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