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Chapter 15: Q5Q (page 616)

A student claimed that the equation for the electric field outside a cube of edge length L, carrying a uniformly distributed charge Q, at a distancex from the center of the cube, was

role="math" localid="1668495301957" E=Qε0Lx1/2

Explain how you know that this cannot be the right equation.

Short Answer

Expert verified

The claimed equation is incorrect because it gives incorrect dimension of electric field.

Step by step solution

01

Identification of given data

The formula for electric field outside a cube of edge length L, carrying a uniformly distributed charge Q, at a distance xfrom the center of the cube is claimed by a student to be

Ec=Qε0Lx1/2

02

Dimensions of electric field and permittivity

The dimension of electric field is

[E]=M1L1T-3I-1 …(i)

The dimension of permittivity is

[ε0]=M-1L-3T4I2 …(ii)

03

Determination of the correctness of the claimed formula

The dimension of the claimed formula, using equation (ii), is

Ec=Q1ε0-1L1x-1/2=I1T1×M-1L-3T4I2-1×L1×L-1/2=ML7/2T-3I-1

This does not match. Hence the formula is incorrect due to incorrect dimension.

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Most popular questions from this chapter

Suppose that the radius of a disk is 21 cm, and the total charge distributed uniformly all over the disk is 5×10-6C. (a) Use the exact result to calculate the electric field 1 mm from the center of the disk. (b) Use the exact result to calculate the electric field 3 mm from the center of the disk. (c) Does the field decrease significantly?

Question: Breakdown field strength for air is roughly . If the electric field is greater than this value, the air becomes a conductor. (a) There is a limit to the amount of charge that you can put on a metal sphere in air. If you slightly exceed this limit, why would breakdown occur, and why would the breakdown occur very near the surface of the sphere, rather than somewhere else? (b) How much excess charge can you put on a metal sphere of radius without causing breakdown in the neighboring air, which would discharge the sphere? (c) How much excess charge can you put on a metal sphere of onlyradius? These results hint at the reason why a highly charged piece of metal tends to spark at places where the radius of curvature is small, or at places where there are sharp points.

An electrostatic dust precipitator that is installed in a factory smokestack includes a straight metal wire of length L=0.8 mthat is charged approximately uniformly with a total charge Q=0.4×10-7C . A speck of coal dust (which is mostly carbon) is near the wire, far from both ends of the wire; the distance from the wire to the speck is d=1.5 cm . Carbon has an atomic mass of 12( 6protons and 6neutrons in the nucleus). A careful measurement of the polarizability of a carbon atom gives the value

∝=1.96×10-40C·mN/C

(a) Calculate the initial acceleration of the speck of coal dust, neglecting gravity. Explain your steps clearly. Your answer must be expressed in terms ofQ,L,d,and ∝. You can use other quantities in your calculations, but your final result must not include them. Don’t put numbers into your calculation until the very end, but then show the numerical calculation that you carry out on your calculator. It is convenient to use the “binomial expansion” that you may have learned in calculus, that(1+ε)n≈1+nεis ε≪1. Note thatcan be negative. (b) If the speck of coal dust were initially twice as far from the charged wire, how much smaller would be the initial acceleration of the speck?

Define “fringe field.”

Consider a capacitor made of two rectangular metal plates of length L and width W, with a very small gap s between the plates. There is a charge +Qon one plate and a charge −Qon the other. Assume that the electric field is nearly uniform throughout the gap region and negligibly small outside. Calculate the attractive force that one plate exerts on the other. Remember that one of the plates doesn’t exert a net force on itself
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