Problem 89 - Fundamentals of Physics

Question

89 SSM For a sinusoidally driven series \(R L C\) circuit, show that over one complete cycle with period \(T\) (a) the energy stored in the capacitor does not change; (b) the energy stored in the inductor does not change; (c) the driving emf device supplies energy \(\left(\frac{1}{2} T\right) \mathscr{E}_{m} I \cos \phi ;\) and (d) the resistor dissipates energy \(\left(\frac{1}{2} T\right) R I^{2}\). (e) Show that the quantities found in (c) and (d) are equal.

Step-by-Step Solution

Answer

The EMF supplies energy that matches the energy dissipated by the resistor over a cycle.

Step 1: Understand the Instantaneous Power

The instantaneous power supplied by a sinusoidal source in an RLC circuit can be expressed through the equation: \( P(t) = I(t) \cdot V(t) = I_m \cdot \sin(\omega t) \cdot \mathscr{E}_m \cdot \sin(\omega t + \phi) \). This equation relates power to the instantaneous current and voltage.

Step 2: Energy in the Capacitor

The energy stored in the capacitor at any time \( t \) is given by \( U_c = \frac{1}{2} C V_c^2 \). Over a complete cycle, the average energy change in the capacitor is zero because the voltage across a capacitor returns to its initial value, thus, \( \Delta U_c = 0 \).

Step 3: Energy in the Inductor

The energy stored in the inductor at any time \( t \) is given by \( U_L = \frac{1}{2} L I^2 \). Similarly, over a complete cycle, the current through an inductor returns to its initial value, therefore, the average energy change in the inductor is also zero, or \( \Delta U_L = 0 \).

Step 4: Energy Supplied by the EMF Source

For the EMF source supplying power, the average power over one cycle is \( \frac{1}{2} \mathscr{E}_m I \cos \phi \). Over a complete cycle \( T \), this corresponds to the energy \( \left(\frac{1}{2} T\right) \mathscr{E}_m I \cos \phi \). This represents the average supply of energy over time from the power source.

Step 5: Energy Dissipated by the Resistor

For a resistor, energy dissipation due to current is given by \( P_R = I^2 R \). The average energy dissipated over one period is \( \left(\frac{1}{2} T\right) R I^2 \), representing the total resistance-related energy loss due to thermal emissions in the resistor.

Step 6: Equate Energy Supplied and Dissipated

To show equality of the EMF supplied energy and energy dissipated by the resistor, note that over a complete cycle, the energy conserved in the circuit results in the time-averaged energy input from the source being equivalent to the energy lost via the resistor's heat dissipation, i.e., \( \left(\frac{1}{2} T\right) \mathscr{E}_m I \cos \phi = \left(\frac{1}{2} T\right) R I^2 \).

Key Concepts

Energy Conservation

Energy conservation in an RLC circuit is all about how energy moves and changes forms without being lost or gained over time. This fundamental principle tells us that the total energy in a closed system like an RLC circuit remains constant, even though it may change forms between electrical energy and heat.

In a sinusoidal-driven RLC circuit, energy transformation is happening all the time. The energy supplied by the source goes into the capacitor, inductor, and resistor but is ultimately conserved. For the whole cycle, the energy changes in the capacitor and inductor are zero.

• The capacitor stores energy temporarily and releases it at different times, but after a full cycle, it ends up back where it started.
• Similarly, the inductor stores energy in its magnetic field when the current flows through it and again returns to its initial energy state after a complete cycle.
• The resistor dissipates energy as heat, which constitutes the circuit's energy loss.

In essence, over a complete cycle, the energy taken from the source equals the energy dissipated as heat by the resistor.

Inductor Energy

The inductor in an RLC circuit is crucial for its ability to store energy in its magnetic field. The energy in the inductor at any point in time can be described by the equation, \( U_L = \frac{1}{2} L I^2 \),where \( L \) is the inductance and \( I \) the current. This equation tells us that the energy stored in the inductor depends on the current flowing through it.

Over one complete cycle in a sinusoidal-driven RLC circuit, the current through the inductor returns to its original value. This cyclical nature means that any energy the inductor gains will be exactly the same amount as it loses throughout the cycle. Hence, the average change in energy of the inductor across one whole cycle is zero. This happens because any energy stored in the inductor's magnetic field is released over the cycle and doesn't build up.

Capacitor Energy

The capacitor in an RLC circuit stores energy in an electric field, and this energy storage can be expressed with the formula:\( U_c = \frac{1}{2} C V_c^2 \),where \( C \) represents the capacitance and \( V_c \) is the voltage across the capacitor. The conservation of energy principle assures us that the energy in the capacitor doesn't change over one full cycle.

During each cycle, the capacitor goes through a process of charging and discharging. The capacitor accumulates energy as it charges but eventually discharges back into the circuit, returning to its initial energy state by the end of the cycle. This repeated cycle of charging and discharging ensures the net change in energy for the capacitor remains zero across the full cycle.

Power Dissipation

Power dissipation in an RLC circuit refers to the energy that is lost in the form of heat, typically within the resistor. This dissipation is a crucial component to consider for analyzing energy conservation.

In this scenario, power dissipation can be calculated using the formula:\( P_R = I^2 R \),where \( R \) represents resistance and \( I \) the current. Over one cycle, the energy dissipated by the resistor is given by:\( \left(\frac{1}{2} T\right) R I^2 \),which highlights that the energy supplied by the EMF matches the energy dissipated as heat by the resistor over that complete period.

• This relationship binds the energy supplied by the driving emf with the thermal energy emitted by the resistor, ensuring that all energy from the source is accounted for as dissipated energy.
• Over one cycle, energy not only flows in and out of the inductor and capacitor but is counterbalanced perfectly by the energy dissipated due to the resistor's operation.

Therefore, in line with energy conservation, the energy input from the source equals the energy lost to resistive heating, and no excess energy accumulates in the system.