LC Circuits: Videos & Practice Problems

An LC circuit consists of an inductor and a capacitor, oscillating energy between electrical and magnetic forms. The maximum charge on the capacitor corresponds to maximum electrical energy, while maximum current through the inductor indicates maximum magnetic energy. The equations governing these energies are ${\frac{1}{2}}^l{I}^2$ for magnetic energy and $Q^2/\frac{2}{C}$ for electrical energy. This oscillation mirrors simple harmonic motion, emphasizing energy conservation in the absence of resistance.

LC Circuits

An LC circuit, consisting of an inductor (L) and a capacitor (C), operates through a continuous cycle of energy transfer between the electric field of the capacitor and the magnetic field of the inductor. Initially, when the capacitor is charged, it holds a maximum amount of charge, and the current is zero. As the capacitor discharges, the current begins to flow, but the inductor resists rapid changes in current, causing the current to increase gradually.

As the current reaches its maximum, the charge on the capacitor becomes zero. At this point, the inductor has stored energy in its magnetic field, and the current starts to decrease as the capacitor begins to charge in the opposite direction. This process continues, with the current oscillating back and forth, similar to simple harmonic motion, where a mass attached to a spring moves back and forth around an equilibrium position.

The oscillation of the LC circuit can be described using sinusoidal functions. The charge \( q(t) \) on the capacitor can be expressed as:

\( q(t) = q_{\text{max}} \cos(\omega t + \phi) \)

Here, \( q_{\text{max}} \) is the maximum charge, \( \omega \) is the angular frequency, and \( \phi \) is the phase angle. The angular frequency \( \omega \) is determined by the inductor and capacitor values, given by:

\( \omega = \sqrt{\frac{1}{LC}} \)

The current \( I(t) \) in the circuit is related to the charge by the equation:

\( I(t) = -\omega q_{\text{max}} \sin(\omega t + \phi) \)

At maximum current, the sine function reaches its peak value of 1 or -1, leading to the maximum current being:

\( I_{\text{max}} = \omega q_{\text{max}} \)

To calculate the maximum current in an LC circuit, one must first determine the angular frequency using the inductance (L) and capacitance (C). For example, if \( L = 4 \, \text{H} \) and \( C = 5 \times 10^{-9} \, \text{F} \), the angular frequency can be calculated as:

\( \omega = \sqrt{\frac{1}{4 \times 5 \times 10^{-9}}} \approx 77071 \, \text{rad/s} \)

Substituting this value into the maximum current equation, along with the maximum charge \( q_{\text{max}} = 2 \times 10^{-4} \, \text{C} \), yields:

\( I_{\text{max}} = 77071 \times 2 \times 10^{-4} \approx 1.41 \, \text{A} \)

In summary, the oscillatory behavior of an LC circuit is characterized by the interplay of electric and magnetic fields, with charge and current described by sinusoidal functions. Understanding these principles is essential for analyzing and designing circuits that utilize inductors and capacitors.

An LC circuit with an inductor of 0.05 H and a capacitor of 35 μF begins with the current of -1A. The capacitor plates have a maximum charge of 2.65mC at any time during the oscillation. What is the phase angle of this oscillation?

• A. 45^o
• B. 15^o
• C. 30^o
• D. 60^o

Answer: (C)

Oscillations in an LC Circuit

In an LC circuit, which consists of an inductor and a capacitor, the behavior of charge transfer is cyclical. When the capacitor is fully charged, the charge begins to oscillate between the two plates of the capacitor as the circuit operates. The time it takes for a fully charged plate to transfer all of its charge to the other plate is essentially half of the complete oscillation cycle, known as the half period.

The period (T) of oscillation in an LC circuit can be determined using the relationship between period, frequency, and angular frequency. The formulas involved are:

1. The period and frequency are inverses:
\( T = \frac{1}{f} \)

2. The angular frequency (\( \omega \)) is related to the frequency by:
\( f = \frac{\omega}{2\pi} \)

3. The angular frequency can also be expressed in terms of inductance (L) and capacitance (C):
\( \omega = \frac{1}{\sqrt{LC}} \)

To find the half period, we first calculate the full period using the angular frequency:

Substituting the expression for angular frequency into the period formula gives us:
\( T = 2\pi \sqrt{LC} \)

Thus, the half period is:
\( \frac{T}{2} = \pi \sqrt{LC} \)

For a specific example, if the inductance \( L \) is 0.05 H and the capacitance \( C \) is 50 mF (which is 0.05 F), we can substitute these values into the half period formula:

Calculating the half period:
\( \frac{T}{2} = \pi \sqrt{0.05 \times 0.05} \)

This results in a half period of approximately 0.157 seconds. Therefore, it takes about 0.157 seconds for the charge to transfer from one plate of the capacitor to the other, completing half of the oscillation cycle. The charge will then take an additional 0.157 seconds to return, thus repeating the cycle.

Energy in an LC Circuit

In an LC circuit, energy oscillates between the capacitor and the inductor, demonstrating the principle of energy conservation in the absence of resistance. The electrical energy stored in the capacitor is found in the electric field between its plates, while the magnetic energy in the inductor arises from the current flowing through a coil of wire, creating a magnetic field. The relationship between these two forms of energy can be expressed through specific equations: the electrical energy \( U_C \) in a capacitor is given by \( U_C = \frac{1}{2} C V^2 \) and the magnetic energy \( U_L \) in an inductor is given by \( U_L = \frac{1}{2} L I^2 \), where \( C \) is capacitance, \( V \) is voltage, \( L \) is inductance, and \( I \) is current.

The energy transfer in an LC circuit occurs in a cyclical manner. Initially, when the capacitor is fully charged, the current is zero, resulting in maximum electrical energy and zero magnetic energy. As the circuit oscillates, the charge on the capacitor decreases, leading to an increase in current and thus an increase in magnetic energy. At the midpoint of the cycle, the capacitor is discharged (zero charge), and the current reaches its maximum, resulting in maximum magnetic energy. The cycle then reverses, with energy oscillating back to the capacitor, demonstrating a continuous exchange between electrical and magnetic energy.

This oscillation can be likened to simple harmonic motion, where potential energy and kinetic energy interchange. In the case of the LC circuit, as one form of energy increases, the other decreases, maintaining a constant total energy throughout the cycle.

To illustrate this concept, consider an example involving an LC circuit with a 0.1 Henry inductor and a 15 nanoFarad capacitor. To find the magnetic energy stored in the inductor after a specific time, we can use the equation for magnetic energy \( U_L = \frac{1}{2} L I^2 \). The current \( I \) at a given time can be determined using the equation \( I(t) = -\omega Q_{\text{max}} \sin(\omega t + \phi) \), where \( \omega = \sqrt{\frac{1}{LC}} \) is the angular frequency, \( Q_{\text{max}} \) is the maximum charge, and \( \phi \) is the phase angle.

In this scenario, if the capacitor starts maximally charged, the phase angle \( \phi \) is zero. By calculating \( \omega \) and substituting the values into the current equation, we can find the current at a specific time, which can then be used to calculate the magnetic energy stored in the inductor. This process highlights the dynamic interplay between electrical and magnetic energies in an LC circuit.

In an oscillating LC circuit in which the capacitance C = 4μF and the maximum voltage across the capacitor V = 1.50V, the maximum current measured across the inductor is 50mA. What is the angular frequency of this LC circuit?

• A. 69444444 rad/s
• B. 8333 rad/s
• C. 1521 rad/s
• D. 0.00012 rad/s

Answer: (B)

Here’s what students ask on this topic:

What is an LC circuit and how does it work?

An LC circuit consists of an inductor (L) and a capacitor (C) connected together. It works by oscillating energy between the capacitor's electric field and the inductor's magnetic field. When the capacitor is fully charged, it has maximum electrical energy and no current flows. As the capacitor discharges, the current increases, creating a magnetic field in the inductor. When the capacitor is fully discharged, the current is at its maximum, and the energy is stored in the inductor's magnetic field. This process then reverses, causing the system to oscillate back and forth, similar to simple harmonic motion.

How is energy conserved in an LC circuit?

In an LC circuit, energy is conserved because there is no resistance to dissipate energy as heat. The total energy oscillates between the capacitor and the inductor. The electrical energy stored in the capacitor is given by $Q^2/2C$, where Q is the charge and C is the capacitance. The magnetic energy stored in the inductor is given by $1/2L{I}^2$, where L is the inductance and I is the current. As the capacitor discharges, its electrical energy decreases while the inductor's magnetic energy increases, and vice versa, maintaining a constant total energy.

What is the relationship between LC circuits and simple harmonic motion?

LC circuits and simple harmonic motion (SHM) are analogous because both systems involve oscillations. In an LC circuit, the charge and current oscillate sinusoidally, similar to how displacement and velocity oscillate in SHM. The angular frequency of an LC circuit is given by $\surd \frac{1}{\mathrm{LC}}$, analogous to the angular frequency in SHM, which depends on the mass and spring constant. Both systems exhibit energy conservation, with energy oscillating between two forms: electrical and magnetic in LC circuits, and potential and kinetic in SHM.

How do you calculate the maximum current in an LC circuit?

The maximum current in an LC circuit can be calculated using the relationship between the maximum charge on the capacitor and the angular frequency. The formula is $I$_max=ωQ_max, where $\omega$ is the angular frequency given by $\surd \frac{1}{\mathrm{LC}}$, and $Q$_max is the maximum charge on the capacitor. By substituting the values of L, C, and $Q$_max, you can find the maximum current.

What are the equations for energy in an LC circuit?

The energy in an LC circuit oscillates between electrical energy in the capacitor and magnetic energy in the inductor. The electrical energy stored in the capacitor is given by $Q^2/2C$, where Q is the charge and C is the capacitance. The magnetic energy stored in the inductor is given by $1/2L{I}^2$, where L is the inductance and I is the current. These equations show how energy is conserved and transferred between the capacitor and inductor during oscillations.