Question

A, B and C are parallel conductors of equal lengths carrying currents $I$, $I$ and $2I$ respectively. Distance between A and B is $x$. Distance between B and C is also $x$. $F_{1}$ is the force exerted by B on A. $F_{2}$ is the force exerted by C on A. Choose the correct answer.}

• A. $F_{1} = 2F_{2}$
• B. $F_{2} = 2F_{1}$
• C. $F_{1} = F_{2}$
• D. $F_{1} = -F_{2}$

Hint:

Force is attractive for currents in same direction, repulsive for opposite directions.

Answer 1

The Correct Option is D

To determine the relationship between the forces \(F_1\) (force exerted by B on A) and \(F_2\) (force exerted by C on A), we use the formula for the magnetic force between two parallel current-carrying conductors:

\(F = \frac{\mu_0 \cdot I_1 \cdot I_2 \cdot l}{2\pi \cdot d}\)

where:

• \(\mu_0\) is the permeability of free space.
• \(I_1\) and \(I_2\) are the currents in the conductors.
• \(l\) is the length of the conductors.
• \(d\) is the distance between the conductors.

Given:

• Current in A, \(I_A = I\)
• Current in B, \(I_B = I\)
• Current in C, \(I_C = 2I\)
• Distance between A and B, \(d = x\)
• Distance between A and C, \(d = 2x\)

Calculate \(F_1\):

\(F_1 = \frac{\mu_0 \cdot I \cdot I \cdot l}{2\pi \cdot x}\)

\(F_1 = \frac{\mu_0 \cdot I^2 \cdot l}{2\pi \cdot x}\)

Calculate \(F_2\):

\(F_2 = \frac{\mu_0 \cdot I \cdot 2I \cdot l}{2\pi \cdot 2x}\)

\(F_2 = \frac{\mu_0 \cdot 2I^2 \cdot l}{4\pi \cdot x}\)

\(F_2 = \frac{\mu_0 \cdot I^2 \cdot l}{2\pi \cdot x}\)

From the calculations, \(F_1 = F_2\) in magnitude. However, since the currents in B and C flow in opposite directions with respect to A, the forces will act in opposite directions. Thus, \(F_1 = -F_2\).

Therefore, the correct answer is \(F_1 = -F_2\).