Question

A rectangular loop of sides $l$ and $b$ and resistance ‘R’ is kept in a region in which the magnetic field varies as $B = B_0 \sin \omega t$. Find the effective value of current that flows in the loop.

Hint:

Effective value $I_{rms}$ represents equivalent DC current that produces the same Joule heating effect in loop resistance $R$ over a full AC time period.

Answer 1

Concept:
• Instantaneous induced current is $i(t) = \frac{e(t)}{R}$.

• Effective or root-mean-square (rms) current for a sinusoidal current $i(t) = I_0 \cos \omega t$ is $I_{rms} = \frac{I_0}{\sqrt{2}}$, where $I_0$ is peak current amplitude.

Step 1: Determine instantaneous current
From Part (i), induced emf is $e(t) = -l b B_0 \omega \cos \omega t$.
Resistance of the loop = $R$.
Using Ohm's law, instantaneous induced current $i(t)$:
\[ i(t) = \frac{e(t)}{R} = -\frac{l b B_0 \omega}{R} \cos \omega t \]

Step 2: Identify peak current amplitude
The peak current amplitude $I_0$ is:
\[ I_0 = \frac{l b B_0 \omega}{R} \]

Step 3: Calculate effective (rms) current
The effective (rms) current value $I_{eff}$ is:
\[ I_{eff} = I_{rms} = \frac{I_0}{\sqrt{2}} \]
Substitute $I_0 = \frac{l b B_0 \omega}{R}$:
\[ I_{eff} = \frac{l b B_0 \omega}{\sqrt{2} R} \]

Step 4: Conclusion
The effective value of current flowing through the loop is $I_{eff} = \frac{l b B_0 \omega}{\sqrt{2} R}$.