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Question
The electric field in a region is directed outward and is proportional to the distance rfrom the origin. Taking the electric potential at the origin to be zero,
Options
it is uniform in the region
it is proportional to r
it is proportional to r2
it increases as one moves away from the origin
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Solution
it is proportional to r2
Given:
E \[\propto\] r and V = 0 at r =0
⇒ E = kr
Also, E = \[\frac{- dV}{dr}\]
\[\therefore V = - \int Edr\]
\[ \Rightarrow V = - \int krdr\]
\[ \Rightarrow V = - k\frac{r^2}{2} + C\]
Using the condition, V = 0 at r = 0, we get C =0.
Therefore,
\[V = - k\frac{r^2}{2}\]
\[ \Rightarrow V \propto r^2\]
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