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The potential in the electric field vari...

The potential in the electric field varies as `V = -a x^2 + b` with respect to x - coordinate, where a and b are constants. Find the charge density `rho (x)` in a space.

Text Solution

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The electric field E is present along x - axis because
`(del V)/(del y) = (del V)/(del z) = 0`
Then, `vec E = -(del V)/(del x) hat i = 2 a x hat i` …(i)
According to Gauss's law, the net flux of `vec E` out of the cubical Gaussion surface is
`E (x + d x) A - E (x) A = (delta q)/(epsilon_0)`
or `(del E)/(del x). A delta x = (delta q)/(epsilon_0)`
or `(del E)/(del x) = (1)/(epsilon_0) ((delta q)/(A delta x)) = (rho (x))/(epsilon_0)`...(ii)
Using Eqs. (i) and (ii),
`(delta)/(delta x) (2 a x) hat i = (rho (x))/(epsilon_0)`
or `rho (x) = 2a epsilon_0`.
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