Dimensional Formula of Electric Potential Explained

This solution details the derivation of the dimensional formula for Electric Potential. We will break down the concept and calculate the dimensions step-by-step.

Understanding Electric Potential

Electric Potential at a point in an electric field is defined as the amount of work required to move a unit positive charge from infinity (or a reference point) to that specific point. It's a measure of the potential energy per unit charge.

Mathematically, it is expressed as:

$$V = \frac{W}{q}$$

Where:

• V represents Electric Potential
• W represents Work done
• q represents Electric Charge

Deriving the Dimensional Formula

To find the dimensional formula of Electric Potential ($[V]$), we need the dimensional formulas of Work ($[W]$) and Charge ($[q]$).

Dimensions of Work (W)

Work is defined as Force multiplied by Distance.

$$W = F \times d$$

We know the dimensional formula for Force ($[F]$) is [M1L1T-2] (Mass × Acceleration).

The dimensions of Distance ($[d]$) is [L1].

Therefore, the dimensions of Work are:

$$[W] = [F] \times [d]$$

$$[W] = [M^1L^1T^{-2}] \times [L^1]$$

$$[W] = [M^1L^2T^{-2}]$$

Dimensions of Electric Charge (q)

Electric Charge is defined as the product of Current and Time.

$$q = I \times t$$

Where:

• I represents Electric Current
• t represents Time

The dimensions of Current ($[I]$) is represented as [I1] (or sometimes [A1] for Ampere).

The dimensions of Time ($[t]$) is [T1].

Therefore, the dimensions of Charge are:

$$[q] = [I^1] \times [T^1]$$

$$[q] = [I^1T^1]$$

Dimensions of Electric Potential (V)

Now, we can combine the dimensions of Work and Charge to find the dimensions of Electric Potential using the formula V = W/q.

$$[V] = \frac{[W]}{[q]}$$

$$[V] = \frac{[M^1L^2T^{-2}]}{[I^1T^1]}$$

To simplify, we move the terms from the denominator to the numerator by changing the sign of their exponents:

$$[V] = [M^1L^2T^{-2} \times I^{-1}T^{-1}]$$

Combining the exponents for Time (T):

$$[V] = [M^1L^2T^{-2-1}I^{-1}]$$

$$[V] = [M^1L^2T^{-3}I^{-1}]$$

Conclusion

The derived dimensional formula for Electric Potential is $M^1L^2T^{-3}I^{-1}$.

Comparing this with the given options:

• Option 1: $M^1L^2T^{-2}I^{-1}$
• Option 2: $M^1L^2T^{-3}I^{-1}$
• Option 3: $M^1L^2T^{-3}I^{-2}$
• Option 4: $M^1L^1T^{-3}I^{-1}$

The calculated dimensional formula matches Option 2.