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Question
A uniform field of 2.0 NC−1 exists in space in the x-direction. (a) Taking the potential at the origin to be zero, write an expression for the potential at a general point (x, y, z). (b) At which point, the potential is 25 V? (c) If the potential at the origin is taken to be 100 V, what will be the expression for the potential at a general point? (d) What will be the potential at the origin if the potential at infinity is taken to be zero? Is it practical to choose the potential at infinity to be zero?
Solution
(a) Given:
Electric field intensity, E = 2 N/C in the x-direction
(a) Potential at the origin = 0
\[V = - E . r\]
\[ = - \left( E_x \hat{i}+ E_y \hat{j }+ E_z \hat{k} \right) . \left( x \hat{i} + y \hat{j } + z \hat{k } \right)\]
\[ \Rightarrow V = - E_x . x = - 2x\]
(b) From the above expression for V, we have
\[(25 - 0) = - 2x\]
\[ \Rightarrow x = \frac{25}{- 2} = - 12 . 5 \] m
(c) If potential at the origin is 100 V, then potential at a general point is given by
\[V - 100 = - 2x\]
\[ \Rightarrow V = 100 - 2x\]
(d) Potential at infinity is given by \[V' - V = - 2x, \]
where V is the potential at the origin.
\[\because V' = 0 , x = \infty , \]
\[V = V' + 2x = \infty\]
It is not practical to take the potential at infinity to be zero because in that case, we have to take the potential at origin to be infinity and the calculations will become practically impossible.789
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