Question

A, B and C are three parallel conductors of equal lengths carrying currents $I$, $I$ and $2I$ respectively. Distance between A and B is $x$ and that between B and C is also $x$. $F_1$ is the force exerted by conductor B on A. $F_2$ is the force exerted by conductor C on A. Current $I$ in A and $I$ in B are in the same direction and current $2I$ in C is in the opposite direction. Then

• A. $F_1 = F_2$
• B. $F_2 = 2F_1$
• C. $F_1 = 2F_2$
• D. $F_1 = -F_2$

Hint:

Notice that for wire C, both the current doubles ($2I$) and the distance doubles ($2x$). Since force is directly proportional to current but inversely proportional to distance ($F \propto \frac{I}{r}$), these two changes cancel out perfectly, meaning the magnitude of the force remains completely unchanged!

Answer 1

The Correct Option is D

Step 1: The arrangement.
Three straight wires A, B and C sit in a line. A and B carry current $I$, and C carries $2I$. The gap A to B is $x$ and the gap B to C is also $x$, so A to C is $2x$. We compare $F_1$ (force on A from B) with $F_2$ (force on A from C).
Step 2: The force law between wires.
Two parallel wires push or pull each other. The force per unit length is \[ F = \frac{\mu_0 I_a I_b L}{2\pi r} \] Same direction currents attract; opposite direction currents repel.
Step 3: Direction of each force on A.
A and B carry current the same way, so they attract; $F_1$ pulls A toward B. A and C carry current the opposite way, so they repel; $F_2$ pushes A away from C, which is the opposite side. So $F_1$ and $F_2$ point in opposite directions.
Step 4: Size of $F_1$.
For A and B, both currents $I$, distance $x$: \[ F_1 = \frac{\mu_0\,I\cdot I\,L}{2\pi x} = \frac{\mu_0 I^2 L}{2\pi x} \]
Step 5: Size of $F_2$.
For A and C, currents $I$ and $2I$, distance $2x$: \[ F_2 = \frac{\mu_0\,I\cdot 2I\,L}{2\pi (2x)} = \frac{\mu_0 I^2 L}{2\pi x} \] The bigger current and the bigger distance cancel out, so the sizes are equal.
Step 6: Combine size and direction.
Equal sizes but opposite directions means \[ F_1 = -F_2 \] This is option (4). \[ \boxed{F_1 = -F_2} \]