Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for $V = 5\sin(100t)$ The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is ___________ $\mu\text{F}$. 
To find the capacitance \(C\) in the given LCR circuit, follow these steps:
The maximum current \(I_{\text{max}}\) is given as \(50 \, \text{mA} = 0.05 \, \text{A}\).
The voltage across the circuit is \(V = 5 \sin(100t)\), so the peak voltage \(V_0 = 5 \, \text{V}\).
The angular frequency \(\omega\) is \(100 \, \text{rad/s}\).
For maximum current, the circuit is at resonance: \(Z = \frac{V_0}{I_{\text{max}}}\), where \(Z\) is the impedance. So, \(Z = \frac{5}{0.05} = 100 \, \Omega\).
At resonance in an LCR circuit, \(Z = R\), meaning \(R = 100 \, \Omega\).
The resonance condition is \(\omega L = \frac{1}{\omega C}\), where \(L = 2 \, \text{H}\).
Thus, \(\omega^2 = \frac{1}{LC}\). Substituting the known values, \(100^2 = \frac{1}{2C}\).
Solve for \(C\):
\(C = \frac{1}{2 \times 10000} = \frac{1}{20000} = 50 \times 10^{-6} \, \text{F} = 50 \, \mu\text{F}\).
The calculated capacitance \(C = 50 \, \mu\text{F}\), which falls within the expected range of [50, 50].
Similar Questions
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Important Questions from Electricity and Magnetism
- A short bar magnet placed with its axis at $30^\circ$ with an external field of 800 Gauss, experiences a torque of 0.016 N.m. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \text{ J}$. The value of $\alpha$ is ______.
- An electromagnetic wave of frequency $100\text{ MHz}$ propagates through a medium of conductivity, $\sigma = 10\text{ mho/m}$. The ratio of maximum conduction current density to maximum displacement current density is __________.
$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$ - An inductor stores $16\text{ J}$ of magnetic field energy and dissipates $32\text{ W}$ of thermal energy due to its resistance when an a.c. current of $2\text{ A}$ (rms) and frequency $50\text{ Hz}$ flows through it. The ratio of inductive reactance to its resistance is _________. ($\pi = 3.14$)
- A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^\circ$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ___________ mV. (Take $g = 10\text{ m/s}^2$)
The space between the plates of a parallel plate capacitor of capacitance C (without any dielectric) is now filled with three dielectric slabs of dielectric constants $K_1 = 2$, $K_2 = 3$ and $K_3 = 5$ (as shown in figure). If new capacitance is $\frac{n}{3}C$ then the value of $n$ is ___________.
