Question

What are the Faraday\'s laws of electromagnetic induction? The magnetic flux linked with a coil of resistance \(2\ \Omega\) is \[ \phi = (5t^{3} - 100t + 300)\ \text{Weber} \] Find out: (i) induced e.m.f. produced in the coil at \(t = 2\) seconds (ii) induced current in the coil.
OR
Explain the conditions required for the resonance in series LCR alternating circuit by the diagram. On which factors does the resonant frequency depend?

Hint:

Induced e.m.f. is \(\varepsilon=-d\phi/dt=100-15t^{2}\); put \(t=2\) then \(I=\varepsilon/R\). For the OR part, resonance needs \(X_L=X_C\), giving \(f_0=1/(2\pi\sqrt{LC})\), which depends on L and C only.

Answer 1

Option 1: Faraday\'s laws and the induced emf / current

Step 1 (Faraday\'s first law): Whenever the magnetic flux linked with a closed circuit changes, an e.m.f. is induced in it, and it lasts only as long as the flux keeps changing.
Step 2 (Faraday\'s second law): The magnitude of the induced e.m.f. equals the rate of change of magnetic flux linkage: \[ \varepsilon = -\frac{d\phi}{dt} \]The minus sign is Lenz\'s law, showing the induced e.m.f. opposes the change producing it.
Step 3 (Differentiate the given flux): With \( \phi = 5t^{3} - 100t + 300 \), \[ \varepsilon = -\frac{d}{dt}\left(5t^{3} - 100t + 300\right) = -\left(15t^{2} - 100\right) = 100 - 15t^{2} \]Step 4 (Evaluate at t = 2 s): \( \varepsilon = 100 - 15(2)^{2} = 100 - 15\times4 = 100 - 60 = 40\ \text{V}. \)
Step 5 (Induced current): By Ohm\'s law \( I = \dfrac{\varepsilon}{R} = \dfrac{40}{2} = 20\ \text{A}. \)
\[\boxed{\varepsilon = 40\ \text{V}, \qquad I = 20\ \text{A}}\]

Option 2: Resonance in a series LCR AC circuit

Step 1 (Reactances): In a series LCR circuit fed by \( V = V_0\sin\omega t \), the inductor offers inductive reactance \( X_L = \omega L \) and the capacitor offers capacitive reactance \( X_C = \dfrac{1}{\omega C} \).
Step 2 (Impedance): The total opposition is \[ Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \]Step 3 (Condition for resonance): Resonance occurs when the inductive and capacitive reactances become equal, \( X_L = X_C \). Then the net reactance is zero, so \( Z = R \) is minimum, the current \( I = V_0/Z \) is maximum, and the current is exactly in phase with the applied voltage (power factor = 1).
Step 4 (Resonant frequency): Putting \( X_L = X_C \): \[ \omega L = \frac{1}{\omega C} \Rightarrow \omega^{2} = \frac{1}{LC} \Rightarrow \omega_0 = \frac{1}{\sqrt{LC}} \]\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]Step 5 (Diagram and dependence): A plot of current \(I\) against frequency \(f\) rises to a sharp peak at \(f = f_0\) and falls off on both sides. The resonant frequency depends only on the inductance \(L\) and the capacitance \(C\); it is independent of the resistance \(R\) (R only decides the sharpness/height of the peak).
\[\boxed{f_0 = \dfrac{1}{2\pi\sqrt{LC}}\ \text{(depends on } L \text{ and } C \text{ only)}}\]