Question

A right-angled triangle ABC, made from a metallic wire, moves at a uniform speed of 0 m/s in its plane as shown in the figure. A uniform magnetic field \( B = 0.5 \, \text{T} \) exists in the perpendicular direction to the plane. Find the induced emf in the segments BC, AC, and AB.

Hint:

The induced emf in a conductor moving through a magnetic field depends on the length of the conductor, its velocity, and the strength of the magnetic field.

Answer 1

Step 1: Understanding the Problem.
We are given a right-angled triangle \( \triangle ABC \) with:
- \( AB = 3 \, \text{m} \),
- \( BC = 4 \, \text{m} \),
- The velocity of the wire is \( v = 0 \, \text{m/s} \),
- The magnetic field is \( B = 0.5 \, \text{T} \),
- The magnetic field is perpendicular to the plane of the wire.
We need to find the induced emf in the three segments: \( BC \), \( AC \), and \( AB \).
Step 2: Formula for Induced EMF.
The induced emf in a moving conductor of length \( L \) moving with a velocity \( v \) in a magnetic field \( B \) is given by the formula: \[ \epsilon = B L v \] Where:
- \( \epsilon \) is the induced emf,
- \( B \) is the magnetic field,
- \( L \) is the length of the conductor,
- \( v \) is the velocity of the conductor.
Step 3: Calculating Induced EMF in Segment BC.
For the segment \( BC \), the length is \( L_{BC} = 4 \, \text{m} \). Substitute the given values into the formula: \[ \epsilon_{BC} = B \times L_{BC} \times v = 0.5 \times 4 \times 2 = 4 \, \text{V} \]
Step 4: Calculating Induced EMF in Segment AC.
For the segment \( AC \), the length is \( L_{AC} = 5 \, \text{m} \) (using the Pythagoras theorem, \( AC = \sqrt{AB^2 + BC^2} = \sqrt{3^2 + 4^2} = 5 \, \text{m} \)).
Substitute the given values into the formula: \[ \epsilon_{AC} = B \times L_{AC} \times v = 0.5 \times 5 \times 2 = 5 \, \text{V} \]
Step 5: Calculating Induced EMF in Segment AB.
For the segment \( AB \), the length is \( L_{AB} = 3 \, \text{m} \).
Substitute the given values into the formula: \[ \epsilon_{AB} = B \times L_{AB} \times v = 0.5 \times 3 \times 2 = 3 \, \text{V} \]
Final Answer:
The induced emf in the segments are:
- \( \epsilon_{BC} = 4 \, \text{V} \),
- \( \epsilon_{AC} = 5 \, \text{V} \),
- \( \epsilon_{AB} = 3 \, \text{V} \).