IIT JEE , AIEEE Forum - Physics , Electricity

Conjurer

The plastic rod of length L in fig has the nonuniform linear charge density ,lambda, = cx , where c is a postive constant.

a) With V =0 at infinity, find the electric potential at point P on the y axis, a distance b from one end of the rod.

b)From that result, find the electric field component E_y at P.

c) Why cannot the field component E_x at P be found using the result of (a)

You can just give the raw equations for a) and b) and leave the calculation part but I am particulary interested in knowing the answer for c)

NOTE: Resnick Halliday Walker/Krane is/are the best book(s) for physics.

kaymant

(i) For some $x$ lying between $a$ and $a+L$ , take an element of length $\mathrm{d}x$ . Then the charge on this piece $\mathrm{d}q=\lambda \,\mathrm{d}x = c x\,\mathrm{d}x$ . The contribution to the potential at P from the chosen element is

$\mathrm{d}V = \dfrac{1}{4\pi\epsilon_0}\dfrac{\mathrm{d} q}{\sqrt{x^2 + b^2}}=\dfrac{1}{4\pi\epsilon_0}\dfrac{cx\mathrm{d}x}{\sqrt{x^2 + b^2}}$

To find the total potential because of the rod, integrate the above as $x$ varies from $a$ to $a+L$ , to obtain, the potential at P as

$V(P) = \dfrac{c}{4\pi\epsilon_0}\left(\sqrt{(a+L)^2+b^2}-\sqrt{a^2 + b^2}\right)$

(ii) At an arbiraray point $(0,\,y)$ on the $y$ axis, we get the potential as

$V=\dfrac{c}{4\pi\epsilon_0}\left(\sqrt{(a+L)^2+y^2}-\sqrt{a^2 + y^2}\right)$

Use this expression to determine $E_y= -\dfrac{\mathrm{d}\,V}{\mathrm{d}\,y}$ .

(iii) We can not use the above expression to determine $E_x$ since it depends only on $y$ .

Conjurer

Sir can you explain c) a little more in detail?

kaymant

You see, in order to determine E_x at P from the expression for V, we must know, how the potential depends on x. In this particular problem, at any point on the y axis, the expression for the potential does not show any dependence on x. So we cannot determine the E_x on the y axis by using the expression for the potential obtained for the points along the y axis. However, if the potential is determined for any arbitrary point (x,y), Then, we can determine the E_x andE_y at all points on the plane.

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