## A rod of length L has a charge Q uniformly distributed along its length. The rod lies along the y axis with one end at the origin. (a) Find

Question

A rod of length L has a charge Q uniformly distributed along its length. The rod lies along the y axis with one end at the origin. (a) Find an expression for the electric potential as a function of position along the x axis.

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3 years 2021-08-22T01:19:41+00:00 1 Answers 7 views 0

## Answers ( )

V = k Q/l  ln [(l +√(l² + x²)) / x]

Explanation:

The electrical potential for a continuous distribution of charges is

V = k ∫ dq / r

Let’s apply this expression to our case, define a linear charge density for the bar

λ = dq / dy

dq = λ dy

The distance from a point on the bar to the x-axis is

r = √ (x² + y²)

Let’s replace

V = K ∫ λ dy /√ (x² + y²)

We integrate

V = k λ ln (y + √ (x² + y²))

Let’s evaluate between y = 0 and y = l

V = k λ [ ln (l +√(x² + l²) – ln x]

We substitute the linear density

V = k Q/l  ln [(l +√(l² + x²)) / x]