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Compare the energy storage in a capacitor and an inductor.
Capacitor: Stores energy in an electric field. | Inductor: Stores energy in a magnetic field.
What are the differences between angular frequency ($\omega$) and frequency ($f$)?
Angular Frequency ($\omega$): Measured in rad/s, represents the rate of change of the phase. | Frequency ($f$): Measured in Hz, represents the number of cycles per second. Relationship: $\omega = 2\pi f$.
Compare the role of the inductor and capacitor in an LC circuit.
Inductor: Opposes changes in current, acting as the restoring force. | Capacitor: Resists changes in voltage, providing the inertia.
Compare the energy equations for a capacitor and inductor at their maximum energy storage.
Capacitor: $U_C = \frac{1}{2}CV_{max}^2$ | Inductor: $U_L = \frac{1}{2}LI_{max}^2$
Compare the effect of increasing L versus increasing C on the angular frequency of an LC circuit.
Increasing L: Decreases the angular frequency ($\omega$). | Increasing C: Decreases the angular frequency ($\omega$).
What happens when the capacitor in an LC circuit is fully charged?
All energy is stored as electric potential energy in the capacitor ($U_C = \frac{1}{2}CV_{max}^2$), and the current is zero.
What happens when the inductor in an LC circuit is fully energized?
All energy is stored in the inductor's magnetic field ($U_L = \frac{1}{2}LI_{max}^2$), and the voltage across the capacitor is zero.
What is the effect of increasing inductance (L) on the period of oscillation (T) in an LC circuit?
Increasing inductance increases the period of oscillation (T).
What is the effect of increasing capacitance (C) on the period of oscillation (T) in an LC circuit?
Increasing capacitance increases the period of oscillation (T).
What happens if a resistor is added in series to an LC circuit?
The oscillations will be damped, and the energy will gradually be dissipated as heat in the resistor, causing the amplitude of the oscillations to decrease over time.
Describe the energy transfer process in an LC circuit.
Energy oscillates between the capacitor (electric field) and the inductor (magnetic field). Capacitor discharges, transferring energy to the inductor. The inductor then discharges, transferring energy back to the capacitor, repeating the cycle.
How do you calculate the maximum current ($I_{max}$) in an LC circuit?
1. Equate the initial energy in the capacitor to the maximum energy in the inductor: $\frac{1}{2}CV_{max}^2 = \frac{1}{2}LI_{max}^2$. 2. Solve for $I_{max}$: $I_{max} = V_{max}\sqrt{\frac{C}{L}}$
What are the steps to deriving the angular frequency ($\omega$) of an LC circuit?
1. Start with Kirchhoff's loop rule: $L\frac{dI}{dt} + \frac{q}{C} = 0$. 2. Substitute $I = \frac{dq}{dt}$: $\frac{d^{2}q}{dt^{2}} = -\frac{1}{LC}q$. 3. Recognize the SHM equation form: $\frac{d^{2}x}{dt^{2}} = -\omega^2x$. 4. Identify $\omega^2 = \frac{1}{LC}$, therefore $\omega = \frac{1}{\sqrt{LC}}$
Explain how to find the charge on the capacitor as a function of time.
1. Recognize that the charge oscillates with SHM. 2. Use the general form: $q(t) = q_{max}\cos(\omega t + \phi)$. 3. Determine $q_{max}$ (maximum charge). 4. Determine $\omega$ (angular frequency). 5. Determine $\phi$ (phase constant, often 0 if the capacitor is initially fully charged).
What is the process of solving for the initial energy stored in a capacitor?
1. Identify the capacitance (C) and voltage (V) across the capacitor. 2. Use the formula: $U_C = \frac{1}{2}CV^2$. 3. Plug in the values of C and V to calculate $U_C$.