If the electric potential is \(V = 500\) volts at the point \((10,\,20)\) and the electric field is given by
\[
\vec{E} = 10x\,\hat{i} + 5y\,\hat{j}\ \text{N/C},
\]
find the potential at the origin.
Show Hint
If electric field components depend only on coordinates, first find the potential function using
\(\vec{E} = -\nabla V\), then apply the given boundary condition to determine the constant.
Concept:
Electric field and electric potential are related by:
\[
\vec{E} = -\nabla V
\]
or,
\[
dV = -\vec{E}\cdot d\vec{r}
\]
If the electric field is conservative (as here, since it depends only on position), a scalar potential function \(V(x,y)\) exists.
Step 1: Determine the potential function.
Given:
\[
\vec{E} = 10x\,\hat{i} + 5y\,\hat{j}
\]
From \(\vec{E} = -\nabla V\):
\[
\frac{\partial V}{\partial x} = -10x
\Rightarrow V = -5x^2 + f(y)
\]
\[
\frac{\partial V}{\partial y} = -5y
\Rightarrow f'(y) = -5y
\Rightarrow f(y) = -\frac{5}{2}y^2 + C
\]
Thus,
\[
V(x,y) = -5x^2 - \frac{5}{2}y^2 + C
\]
Step 2: Use the given potential at \((10,\,20)\).
\[
500 = -5(10)^2 - \frac{5}{2}(20)^2 + C
\]
\[
500 = -500 -1000 + C
\]
\[
C = 2000
\]
Step 3: Find the potential at the origin.
At \((0,0)\):
\[
V(0,0) = C = 2000\ \text{V}
\]
\[
\boxed{V_{\text{origin}} = 2000\ \text{volt}}
\]
