Understanding the Dimensions of EMF

The question asks for the dimensions of Electromotive Force (EMF). EMF is a measure of the energy supplied by a source (like a battery) per unit electric charge that flows through it. It is essentially equivalent to potential difference or voltage in terms of units and dimensions.

To find the dimensions of EMF, we can use the formula relating potential difference (V) to work (W) and charge (Q):

\(V = \frac{W}{Q}\)

Now, let's determine the dimensions of Work (W) and Charge (Q).

Dimensions of Work (W)

Work is defined as Force multiplied by Distance. The dimensions of Force are derived from Newton's second law, \(F = ma\), where m is mass and a is acceleration.

• Dimensions of Mass (m): \([M]\)
• Dimensions of Acceleration (a): Acceleration is change in velocity per unit time. Velocity has dimensions \([LT^{-1}]\). So, acceleration has dimensions \(\frac{[LT^{-1}]}{[T]} = [LT^{-2}]\).

Therefore, the dimensions of Force are \([F] = [M] \times [LT^{-2}] = [MLT^{-2}]\).

Work is Force times Distance. The dimensions of Distance are \([L]\).

So, the dimensions of Work are \([W] = [F] \times [Distance] = [MLT^{-2}] \times [L] = [ML^2T^{-2}]\).

Dimensions of Charge (Q)

Electric charge is related to electric current (I) and time (t) by the definition of current: \(I = \frac{Q}{t}\). Therefore, \(Q = It\).

• Dimensions of Electric Current (I): The dimension of current is taken as a fundamental dimension, denoted by \([A]\) (for Ampere).
• Dimensions of Time (t): The dimension of time is a fundamental dimension, denoted by \([T]\).

So, the dimensions of Charge are \([Q] = [I] \times [t] = [A] \times [T] = [AT]\).

Calculating Dimensions of EMF

Now we can substitute the dimensions of Work and Charge into the formula for EMF (or Voltage):

Dimensions of EMF \(= \frac{\text{Dimensions of Work}}{\text{Dimensions of Charge}}\)

Dimensions of EMF \(= \frac{[ML^2T^{-2}]}{[AT]}\)

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

Dimensions of EMF \(= [ML^2T^{-2}] \times [A^{-1}T^{-1}]\) Dimensions of EMF \(= [ML^2T^{-2-1}A^{-1}]\) Dimensions of EMF \(= [ML^2T^{-3}A^{-1}]\)

So, the dimensions of EMF are \(ML^2T^{-3}A^{-1}\).

Let's compare this with the given options:

• Option 1: \(ML^2T^{-3}A^{-1}\)
• Option 2: \(ML^2T^2A^3\)
• Option 3: \(M^{-1}T^3\)
• Option 4: \(ML^3T^1A^3\)

The calculated dimensions match Option 1.

Revision Table: Common Physical Quantities and Their Dimensions

Additional Information on EMF and Related Concepts

Electromotive Force (EMF): While often called a "force," EMF is not a mechanical force. It is the energy per unit charge created by a source, like a battery or a generator, to drive current through a circuit. It's the maximum potential difference between the terminals of the source when no current is flowing.

Potential Difference (Voltage): Potential difference is the work done per unit charge to move a charge between two points in an electric field. EMF is a specific type of potential difference related to the source itself.

Electric Field (E): The dimensions of the electric field can be found using the formula \(F = qE\), where F is force and q is charge. Dimensions of E = \(\frac{[F]}{[q]} = \frac{[MLT^{-2}]}{[AT]} = [MLT^{-3}A^{-1}]\). Notice the similarity in the time and current components with EMF, but with one power of Length less.

Understanding dimensions helps verify equations and understand the physical nature of quantities.