An equilateral triangular loop having a resistance R and length of each side l is placed in a magnetic field which is varying at   The induced current in the loop will be    

Electromagnetic Induction in Triangular Loop

Solution:

• Calculate the area of the equilateral triangle:

A = (√3/4) × l²

For an equilateral triangle with side length l, the area is derived using the standard formula.

• Determine the magnetic flux:

Φ = B × A = B × (√3/4) × l²

Where Φ is the magnetic flux through the loop and B is the magnetic field.

• Find the rate of change of magnetic flux:

dΦ/dt = A × (dB/dt) 

dΦ/dt = (√3/4) × l² × 1 

dΦ/dt = (√3/4) × l² T·m²/s

• Apply Faraday's Law of Electromagnetic Induction:

Faraday's Law:

ε = |dΦ/dt|

The induced electromotive force (emf) is equal to the rate of change of magnetic flux.

ε = (√3/4) × l² volts

• Calculate the induced current using Ohm's Law:

Ohm's Law:

I = ε / R

I = [(√3/4) × l²] / R 

I = (√3 × l²) / (4R)

Key Concepts Used:

• Faraday's Law: The induced emf in a closed loop equals the negative rate of change of magnetic flux through the loop.
• Ohm's Law: Current equals voltage (emf) divided by resistance.
• Magnetic Flux: Product of magnetic field and area perpendicular to the field.
• Lenz's Law: The direction of induced current opposes the change in magnetic flux (indicated by negative sign in Faraday's law).

Final Answer

I = (√3 × l²) / (4R)

Or equivalently:

I = (√3 l²) / (4R) amperes

Physical Interpretation:

The induced current is:

• Directly proportional to the square of the side length (l²), as larger area means more flux change.
• Inversely proportional to the resistance (R), as higher resistance opposes current flow.
• Proportional to the rate of change of magnetic field (given as 1 T/s).
• The factor √3/4 comes from the geometry of the equilateral triangle.