The question asks for the resonance frequency ($f_0$) of a series AC circuit containing a resistor (R), capacitor (C), and inductor (L).
Resonance Frequency Formula
The resonance frequency in an RLC circuit is the frequency at which the inductive reactance equals the capacitive reactance. It is independent of the resistance and is calculated using the formula:
$f_0 = \frac{1}{2\pi\sqrt{LC}}$
Circuit Component Values
Given values are:
- Inductance, $L = 1 \text{ mH} = 1 \times 10^{-3} \text{ H}$
- Capacitance, $C = 0.1 \ \mu\text{F} = 0.1 \times 10^{-6} \text{ F} = 1 \times 10^{-7} \text{ F}$
- Resistance, $R = 1 \text{ k}\Omega = 1000 \ \Omega$ (Note: Resistance does not affect the resonance frequency itself).
Calculation Steps
- Substitute the values of L and C into the resonance frequency formula:
$f_0 = \frac{1}{2\pi\sqrt{(1 \times 10^{-3} \text{ H}) \times (1 \times 10^{-7} \text{ F})}}$.
- Simplify the term under the square root:
$LC = 1 \times 10^{-10} \text{ H}\cdot\text{F}$
$\sqrt{LC} = \sqrt{1 \times 10^{-10}} = 1 \times 10^{-5} \text{ s}$
- Calculate the resonance frequency in Hertz (Hz):
$f_0 = \frac{1}{2\pi \times (1 \times 10^{-5})} = \frac{10^5}{2\pi} \text{ Hz}$
Using $\pi \approx 3.14159$,
$f_0 \approx \frac{100000}{2 \times 3.14159} \approx \frac{100000}{6.28318} \approx 15915.5 \text{ Hz}$
- Convert the frequency to kilohertz (kHz):
$f_0 \approx 15.9155 \text{ kHz}$
Conclusion
The calculated resonance frequency is approximately $15.9 \text{ kHz}$.
Similar Questions
- A galvanometer of resistance $100 \ \Omega$ gives full scale deflection for a current of $1 \text{ mA}$. It is converted into an ammeter of range $0 - 10 \text{ A}$. The shunt required is :
In the circuit shown below, the voltage appearing across the diode D will be of the form :
A 100-turn closely wound circular coil of radius $10 \text{ cm}$ has a magnetic field of $3.14 \times 10^{-3} \text{ T}$ at its centre. The current passing through the coil, and the magnitude of the magnetic moment of this coil are, respectively :
(Take $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$)The current $I$ in the circuit shown below is :
(All diodes are ideal and identical.
- Consider two uncharged capacitors of equal capacitance $200 \text{ pF}$. One of them is charged by a $100 \text{ V}$ supply and disconnected. Now this capacitor is connected to the uncharged capacitor. The amount of electrostatic energy lost in the process is :
- Two statements are given below :
A. When the forward bias voltage across a p-n junction diode increases above a certain threshold voltage, the diode current increases significantly.
B. This current is called reverse saturation current.
Choose the correct answer from the options given below : - A rectangular wire loop of sides $8 \text{ cm}$ and $3 \text{ cm}$ with a small cut, is moving out of a region of uniform magnetic field of magnitude $0.3 \text{ T}$ directed normal to the plane of the loop. The emf developed across the cut, if the velocity of the loop is $2 \text{ cm s}^{-1}$ in a direction normal to the shorter side of the loop, will be :
- A resistor is connected to a battery of $12 \text{ V}$ emf and internal resistance $2 \ \Omega$. If the current in the circuit is $0.6 \text{ A}$, the terminal voltage of the battery is :
The figure given below shows a long straight solid wire of circular cross-section of radius 'a' carrying steady current I. The current I is uniformly distributed across its cross-section. The plot which correctly represents the variation of magnetic field (B) with distance (r) from the axis of the conductor in the region is :
Five capacitors of capacitances $C_1 = C_2 = C_3 = C_4 = 10 \ \mu\text{F}$ and $C_5 = 2.5 \ \mu\text{F}$ are connected as shown along with a battery of $50 \text{ V}$.
The equivalent capacitance and the charges on each capacitor respectively are :
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 $g = 10\text{ m/s}^2$)
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 ___________ $\mu\text{F}$.
