What do you understand by potential gradient ?
Establish a relation between electric field and potential gradient.

Options

A. Electric field is in the direction in which the potential decreases steepest
B. Magnitude of electric field is given by the charge in the magnitude of potential per unit displacement jnormal to the equipotential surface at the point.
C. In the region of strong electrric field, equipotential surfaces are far apart.
D. Both the statements (a) and (b) are correct.

Explanation

### Step-by-Step Solution

**Step 1: Understanding Potential Gradient**
The potential gradient is defined as the rate of change of electric potential (V) with respect to distance (r). Mathematically, it is expressed as:

\[ \text{Potential Gradient} = \frac{dV}{dr} \]

where \(dV\) is the change in electric potential and \(dr\) is the change in distance.

**Step 2: Relation between Electric Field and Potential Gradient**
To establish the relationship between electric field (E) and potential gradient, we consider two equipotential surfaces separated by a small distance \(dr\). Let the potential at one surface be \(V\) and at the other surface be \(V - dV\).

**Step 3: Work Done in Moving a Charge**
When a charge is moved from one equipotential surface to another, work done (dW) against the electric field is given by:

\[ dW = -E \cdot dr \]

The negative sign indicates that work is done against the electric field.

**Step 4: Relating Work Done to Potential Difference**
The work done in moving the charge can also be expressed in terms of potential difference:

\[ dW = V_A - V_B = V - (V - dV) = dV \]

**Step 5: Equating the Two Expressions**
By equating the two expressions for work done, we have:

\[ -E \cdot dr = dV \]

**Step 6: Solving for Electric Field**
Rearranging the equation gives:

\[ E = -\frac{dV}{dr} \]

This shows that the electric field (E) is equal to the negative of the potential gradient.

**Step 7: Conclusion**
Thus, we conclude that:

\[ E = -\frac{dV}{dr} \]

This means that the electric field is directed from regions of higher potential to regions of lower potential, and its magnitude is proportional to the rate of change of potential with respect to distance.