Q79P CALC Positive charge Q is distri... [FREE SOLUTION]

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University Physics with Modern Physics

Hugh D. Young, Roger A. Freedman

Physics

14th edition Edition 路 1596 pages

ISBN: 9780321973610

Chapter 4: Q79P (page 718)

CALC Positive charge Q is distributed uniformly along the x-axis from x = 0 to x = a. A positive point charge q is located on the positive x-axis at x = a + r, a distance r to the right of the end of Q (Fig. P21.79). (a) Calculate the x- and y-components of the electric field produced by the charge distribution Q at points on the positive x-axis where x 7 a. (b) Calculate the force (magnitude and direction) that the charge distribution Q exerts on q. (c) Show that if r W a, the magnitude of the force in part (b) is approximately Qq>4pP0r2. Explain why this result is obtained.

Short Answer

Expert verified

a) x- and y-components of the electric field produced by the charge distribution Q at points on the positive x-axis is $\frac{kQ}{a} [\frac{1}{x - a} - \frac{1}{x}]$

b) the force (magnitude and direction) that the charge distribution Q exerts on q is $\frac{kQ}{a} [\frac{1}{r} - \frac{1}{r + a}]$

c) $F = \frac{kqQ}{r^{2}}$

Step by step solution

Step 1:

a) At any point on the x- axis, the vertical component is zero, and the linear distribution $位$ is given by

$位 = \frac{d Q}{d I} = \frac{Q}{a}$

the horizontal difference component is given by

${dE}_{x} = \frac{办位 (dI)}{r^{3}} = \frac{办位 (dI)}{{(x - I)}^{2}}$ (1)

Integrate (1) to get

$E = E_{x} = {鈭�}_{0}^{a} \frac{办位 (dI)}{r^{3}} = \frac{办位 (dI)}{{(x - I)}^{2}} = - 办位 [\frac{- 1}{x - a} - \frac{1}{x}]$

Hence the direction is in + i

Step 2:

b) The force is repulsive force and x=r + a

$F = q E = \frac{k q Q}{a} (\frac{1}{r} - \frac{1}{r + a})$

Therefore the force is repulsive so the direction is in + i

Step 3:

The force F may be written in the form

$F = \frac{kqQ}{a} \frac{(r + a) - r)}{r (r + a)} = \frac{kqQ}{r^{2}}$

$a) E = \frac{k q}{a} (\frac{1}{x - 1} - \frac{1}{x}), i n + i b) F = \frac{k q Q}{a} (\frac{1}{r} - \frac{1}{r + a}), i n + i c) F = \frac{k q Q}{r^{2}}$