Question

A current carrying rectangular loop $P Q R S$ is made of uniform wire The length $P R=Q S=5 \,cm$ and $P Q=R S=100\, cm$ If ammeter current reading changes from $I$ to $2 I$, the ratio of magnetic forces per unit length on the wire $P Q$ due to wire $R S$ in the two cases respectively $\left(f_{P Q}^I: f_{P Q}^{2 I}\right)$ is:

• A. $1: 5$
• B. $1: 3$
• C. $1: 2$
• D. $1: 4$

Answer 1

The Correct Option is D

The problem involves a rectangular loop PQRS made of a uniform wire with current flowing through it. We need to find the ratio of the magnetic forces per unit length on the wire PQ due to the change in current.

Given: - Length \( PR = QS = 5 \, \text{cm} \) - Length \( PQ = RS = 100 \, \text{cm} \)
The current changes from \( I \) to \( 2I \).
We are asked to find the ratio of magnetic forces per unit length on the wire PQ in the two cases, denoted as \( f_I \) and \( f_{2I} \), respectively.

Magnetic Force Formula: The magnetic force per unit length on a straight current-carrying conductor in a magnetic field is given by the formula: \[ f = B \cdot I \cdot L \] Where: 
\( B \) is the magnetic field strength,
\( I \) is the current,
\( L \) is the length of the wire in the magnetic field.

Step 1: Magnetic field due to the current on the wire PQ can be calculated using Ampere's law, but since the wire is rectangular and the length of the segments PR and QS are equal, we focus on the effect of current in the longer segments PQ and RS.

Step 2: Since the current increases from \( I \) to \( 2I \), the magnetic force will change quadratically with the current, as shown by the relation: \[ f_{2I} = 4 \cdot f_I \] This is because force is proportional to the square of the current, i.e., \( f \propto I^2 \).

Step 3: Therefore, the ratio of the magnetic forces per unit length on the wire PQ due to the current \( I \) and \( 2I \) is:

\[ f_I : f_{2I} = 1 : 4 \]

The correct answer is: (D): 1:4