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Chapter 20: Problem 55

A thin rod lies on the \(x\) -axis between \(x=0\) and \(x=L\) and carries total charge \(Q\) distributed uniformly over its length. Show that the electric field strength for \(x>L\) is given by \(E=k Q /[x(x-L)]\)

Short Answer

Expert verified
Therefore, for \(x > L\), the electric field strength is indeed given by \(E = k Q /[x (x - L)]\), where \(k\) is Coulomb's constant.

Step by step solution

01

Define the charge distribution

First, describe the charge on the rod as a continuous distribution. Define the charge density \(\lambda = Q/L\), where \(Q\) is the total charge and \(L\) is the length of the rod.
02

Consider an infinitesimal element

Consider an infinitesimal element dx at a distance x from the origin (0 on the x-axis). The charge on this infinitesimal element can be given as \(dq = \lambda dx = (Q/L) dx\).
03

Find the contribution to the electric field

The contribution dE to the electric field at a point on the x-axis beyond the rod (\(x > L\)) due to this charge is given by Coulomb's Law expressed in differential form, \(dE = k dq / r^2\), where \(r = x - x'\), \(x'\) being the variable of integration indicating position of the infinitesimal charge along the rod.
04

Integrate to find the total field

Integrate this from 0 to \(L\) to get the total electric field, \(E = \int_0^L k (Q/L) dx / (x - x')^2\). The integral, when evaluated yields \(E = k Q /[x (x - L)]\), confirming the given expression.

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Most popular questions from this chapter

A straight wire 10 m long carries \(25 \mu C\) distributed uniformly over its length. (a) What's the line charge density on the wire? Find the electric field strength (b) \(15 \mathrm{cm}\) from the wire axis, not near either end, and (c) 350 m from the wire. Make suitable approximations in both cases.

Find the line charge density on a long wire if a 6.8 - \(\mu \mathrm{g}\) particle carrying 2.1 nC describes a circular orbit about the wire with speed \(280 \mathrm{m} / \mathrm{s}\)

A thin rod extends along the \(x\) -axis from \(x=0\) to \(x=L\) and carries line charge density \(\lambda=\lambda_{0}(x / L)^{2},\) where \(\lambda_{0}\) is a constant. Find the electric field at \(x=-L\).

A dipole with dipole moment \(1.5 \mathrm{nC} \cdot \mathrm{m}\) is oriented at \(30^{\circ}\) to a 4.0-MN/C electric field. Find (a) the magnitude of the torque on the dipole and (b) the work required to rotate the dipole until it's antiparallel to the field.

Find the magnitude of the electric field due to a charged ring of radius \(a\) and total charge \(Q\) on the ring axis at distance \(a\) from the ring's center.
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