Relation between electric field and potential is considered one of the most asked concept.
55 Questions around this concept.
Assume that an electric field exists in space. Then the potential difference VA - VO, where VO is the potential at the origin and VA the potential at x=2 m is :
An electric field is given by kN/C. The potential of the point (1, –2), if potential of the point (2, 4) is taken as zero, is
Two plates are 2 cm apart, a potential difference of 10 volts is applied between them, then the electric field between the plates is:
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If potential $V=-5 x+3 y+\sqrt{15} z$, then electric field $\mathrm{E}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is:
Consider a uniform electric field in the $\hat{z}$ direction. The potential is a constant
a) in all space
b) on the x-y plane for a given z
c) for any y for a given z
d) for any x for a given z
Electric field and potential are related as
$
\vec{E}=-\frac{d V}{d r}
$
Where E is Electric field
And V is Electric potential
And $r$ is the position vector
And Negative sign indicates that in the direction of intensity the potential decreases.
If $\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}$
Then $E_x=\frac{\delta V}{d x}, E_y=\frac{\delta V}{d y}, E_z=\frac{\delta V}{d z}$
where
$
\begin{aligned}
E_x & =-\frac{\partial V}{d x} \quad \text { (partial derivative of } \mathrm{V} \text { w.r.t. x) } \\
E_y & =-\frac{\partial V}{d y} \quad \text { (partial derivative of } \mathrm{V} \text { w.r.t. y) } \\
E_z & =-\frac{\partial V}{d z} \quad \text { (partial derivative of } \mathrm{V} \text { w.r.t. z) }
\end{aligned}
$
Proof-
Let the Electric field at a point $r$ due to a given mass distribution is $E$.
If a test charge $q$ is placed inside a uniform Electric field $E$.
Then force on a charged particle q when it is at r is $\vec{F}=q \vec{E}$ as shown in figure
As the particle is displaced from r to r + dr the
work done by the Electric force on it is
$
d W=\vec{F} \cdot \vec{r}=q \vec{E} \cdot d \vec{r}
$
Electric potential V is defined as the negative of work done by electric force per unit charge
$
d V=-\frac{d W}{q}
$
So Integrating between $r_1$, and $r_2$
We get
$
V\left(\overrightarrow{r_2}\right)-V\left(\overrightarrow{r_1}\right)=\int_{r_1}^{r_2}-\vec{E} \cdot d \vec{r}
$
If $r_1=r_0$, is taken at the reference point, $\mathrm{V}\left(\mathrm{r}_0\right)=0$.
Then the potential $V\left(r_2=r\right)$ at any point $r$ is
$
V(\vec{r})=\int_{r_0}^r-\vec{E} \cdot d \vec{r}
$
in Cartesian coordinates, we can write
$
\vec{E}=E_x \vec{i}+E_y \vec{j}+E_z \vec{k}
$
If $\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}$
Then $d \vec{r}=d x \vec{i}+d y \vec{j}+d z \vec{k}$
So
$
\begin{gathered}
\vec{E} \cdot d \vec{r}=-d V=E_x d x+E_y d y+E_z d z \\
d V=-E_x d x-E_y d y-E_z d z
\end{gathered}
$
If y and z remain constant, $\mathrm{dy}=\mathrm{dz}=0$
$
{ }_{\text {Thus }} E_x=\frac{d V}{d x}
$
Similarly
$
E_y=\frac{d V}{d y}, E_z=\frac{d V}{d z}
$
- When an electric field is a uniform (constant)
As Electric field and potential are related as
$
d V=\int_{r_0}^r-\vec{E} \cdot d \vec{r}
$
and $\mathrm{E}=$ constant then
$
d V=-\vec{E} \int_{r_0}^r d \vec{r}=-\vec{E} d r
$
- If at any region $\mathrm{E}=0$ then $\mathrm{V}=$ constant
- If $\mathrm{V}=0$ then E may or may not be zero.
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