Talha's Physics Academy
Electric Potential and Electric Intensity
Video Lecture
Watch the complete video lecture below to understand electric potential and the derivation of the relation between electric field intensity and potential difference.
Electric Potential and Potential Difference
Electric Potential
Let a very small test charge $q_0$ move from a point $P$ to a point $A$ along any arbitrary path in the electric field as shown in the figure. In order to determine the work done, the path is divided into small elements each of length $\Delta r$. $\Delta r$ is chosen to be so small that it may be regarded as a straight path and the electric field at all of its points remains constant.
The electrostatic force on the charge at point $A$ is given by:
$$F = E q_0$$
Where $E$ is the electric field intensity at $A$ in the direction shown in the figure.
The work done ($\Delta W$) in moving the charge $q_0$ across a small element $\Delta r$ is:
$$\Delta W = F \cdot \Delta r = (E q_0) \Delta r \cos\theta$$
Since the change in potential energy per unit charge defines the potential difference $\Delta V$:
$$\Delta V = \frac{\Delta W}{q_0}$$
Definition: Potential difference between two points in an electric field is defined as the work done in moving a test charge from one point to the other divided by the magnitude of the test charge. Its SI unit is the Volt (V).
Relation Between Electric Field and Potential (Potential Gradient)
Consider two points $a$ and $b$ separated by a very small distance $\Delta s$ on a line of force $AB$, as shown in the figure. The electric field is practically constant over this small distance $\Delta s$. If a test charge $q_0$ is moved from $a$ to $b$, work is done by the electric field on the test charge, which is given by:
$$W = F \cdot \Delta s$$
Since $F = E q_0$:
$$W = (E q_0) \Delta s$$
The work done per unit charge is:
$$\frac{W}{q_0} = E \Delta s$$
Since the work done per unit charge equals the potential difference between points $b$ and $a$ ($\Delta V$):
$$\Delta V = E \Delta s$$
$$E = -\frac{\Delta V}{\Delta s} \quad \text{--- (1)}$$
Potential Gradient: The term $\frac{\Delta V}{\Delta s}$ represents the change of potential per unit distance, which is called the Potential Gradient. The negative sign indicates that the electric field points in the direction of decreasing electric potential.
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