Instantaneous and Average Power in Resistors

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Questions and Answers

What is the significance of rms values in AC circuits?

They define the peak current in the circuit.

They indicate the maximum power that can be handled.

They represent the minimum voltage required for operation.

They simplify the comparison between AC and DC circuits. (correct)

From the provided information, what is the peak voltage given a 220 V rms value?

440 V

311 V (correct)

1.414 V

220 V

Which equation relates AC voltage to AC current in an equivalent way to DC?

V = I^2R

V = P/I

V = IR (correct)

V = P/R

What is the equivalent DC current in terms of rms current?

It generates the same average power loss as AC current. Signup and view all the answers

If a light bulb is rated at 100 W for a 220 V supply, what is the bulb's resistance?

121 Ohms Signup and view all the answers

What is the formula for instantaneous power dissipated in a resistor?

p = i m^2 R sin^2 ωt Signup and view all the answers

What does the bar over a letter signify in the context of average power?

It shows that the value has been averaged over a cycle. Signup and view all the answers

What is the average value of the power over a cycle when expressed as <i^2 R>?

p = i_m^2 R <sin^2 ωt> Signup and view all the answers

Using the trigonometric identity, what is the average value of <sin^2 ωt>?

1/2 Signup and view all the answers

What is the final expression for average power (P) in terms of rms current?

P = i_m^2 R Signup and view all the answers

What does the effective current, or root mean square (rms), signify in alternating current?

It is the constant current that produces the same power as the rms current. Signup and view all the answers

Which historical figure was a strong advocate for alternating current over direct current?

George Westinghouse Signup and view all the answers

What does <cos 2ωt> equal in terms of average value?

0 Signup and view all the answers

What does the variable vm represent in the expression for the ac voltage?

The amplitude of the oscillating potential difference Signup and view all the answers

What is the relationship between ac voltage and current through a resistor according to Kirchhoff’s loop rule?

Voltage equals current times resistance Signup and view all the answers

Who conceived the idea of the rotating magnetic field, essential for alternating current machinery?

Nikola Tesla Signup and view all the answers

Which invention is NOT attributed to Nikola Tesla?

Direct current generator Signup and view all the answers

What does the angular frequency ω correspond to in the context of the given potential difference equation?

The speed at which the sine wave oscillates Signup and view all the answers

Which of the following describes the SI unit for magnetic field named after Nikola Tesla?

Tesla Signup and view all the answers

What is the mathematical expression for current through a resistor when an ac voltage is applied?

i = vm sin ω t / R Signup and view all the answers

Which concept does NOT connect to Nikola Tesla's contributions to electric power?

Simple batteries Signup and view all the answers

What is the effect of having negligible resistance in an inductor when analyzing an AC circuit?

It allows for the application of Kirchhoff's loop rule without additional terms. Signup and view all the answers

In the equation $\frac{di}{dt} v - L = 0$, what does the term $L$ represent?

The self-inductance of the inductor. Signup and view all the answers

How are voltage and current represented in an AC circuit with induction?

As phasors that behave like rotating vectors. Signup and view all the answers

According to Lenz's law, what is the significance of the negative sign in the equation $\frac{di}{dt} v - L = 0$?

It shows that the induced current opposes the change in voltage. Signup and view all the answers

What mathematical role do phasors play in understanding AC circuits?

They simplify the process of adding scalar quantities. Signup and view all the answers

What is the formula for capacitive reactance?

Xc = 1/wC Signup and view all the answers

How does capacitive reactance compare to resistance in terms of current limitation?

It limits current in a similar manner as resistance. Signup and view all the answers

What is the phase difference between current and voltage in a purely capacitive circuit?

Current leads voltage by π/2. Signup and view all the answers

What happens to the average power supplied to a capacitor over a complete cycle?

It is zero. Signup and view all the answers

What occurs when a direct current (dc) source is connected to a capacitor?

The capacitor gets charged and no current flows afterward. Signup and view all the answers

What will happen if the capacitance of the capacitor is reduced when connected to a dc source?

There will be no change to the lamp. Signup and view all the answers

What is the unit of capacitive reactance?

Ohm (Ω) Signup and view all the answers

What is the expression for instantaneous power supplied to the capacitor?

pc = im vm cos(ωt) sin(ωt) Signup and view all the answers

Study Notes

Instantaneous Power in Resistors

Instantaneous power dissipated in a resistor is calculated using the formula ( p = i^2 R ).
For alternating current (AC), this can be expressed as ( p = i_m^2 R \sin^2(\omega t) ).
The average power over a complete cycle is represented as ( \langle p \rangle = \langle i_m^2 R \sin^2(\omega t) \rangle ).

Average Power Calculation

Since ( i_m^2 ) and ( R ) are constants, the average power simplifies to ( \langle p \rangle = i_m^2 R \langle \sin^2(\omega t) \rangle ).
The average value of (\sin^2(\omega t)) over a full cycle is ( \langle \sin^2(\omega t) \rangle = \frac{1}{2} ).
Consequently, average power becomes ( \langle p \rangle = \frac{1}{2} i_m^2 R ).

Root Mean Square (RMS) Current

The RMS (root mean square) current, denoted as ( I_{\text{rms}} ), is defined for AC systems to equate the power with DC systems.
The effective current ( I_{\text{rms}} ) yields the same average power loss as direct current (DC).
AC voltage across a resistor is given by ( v = v_m \sin(\omega t) ), where ( v_m ) is the peak voltage.

Alternating Current and Ohm's Law

Kirchhoff’s loop rule applied to an AC circuit gives ( v_m = i_m R ).
The relationship between AC current and voltage mirrors that of DC: ( V = IR ).
RMS voltage is often indicated in specifications (e.g., household voltage is typically 220V).

Example Problem for a Light Bulb

For a light bulb rated at 100W on a 220V supply:
- Resistance can be calculated as ( R = \frac{V^2}{P} ).
- Peak voltage can be calculated using ( v_m = \sqrt{2} \times V_{\text{rms}} ).
- RMS current through the bulb can be calculated using ( I_{\text{rms}} = \frac{P}{V_{\text{rms}}} ).

Behavior of Inductors in AC Circuits

Inductors exhibit self-inductance ( L ) and follow the equation ( v - L \frac{di}{dt} = 0 ).
Voltage and current in purely inductive AC circuits relate similarly to resistive circuits, modifying through reactance.

Capacitive Reactance

Capacitive reactance is denoted by ( X_c = \frac{1}{\omega C} ).
Current amplitude in a capacitive circuit is found as ( i_m = \frac{v_m}{X_c} ).
Capacitive reactance is proportional to capacitance and inversely proportional to frequency.

Phase Relationship in AC Circuits

In capacitive circuits, the current leads the voltage by ( \frac{\pi}{2} ).
Conversely, in inductive circuits, the current lags the voltage by ( \frac{\pi}{2} ).
Average power through capacitors is zero since no net energy is transferred over a cycle.

Observations with Capacitors and DC Sources

When a DC source charges a capacitor, current flow ceases once charged, resulting in a non-glowing lamp.
Changes in capacitance do not alter the current flow in a DC circuit once the capacitor is charged.

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Description

Explore the concepts of instantaneous and average power in resistors, particularly in AC circuits. This quiz covers formulas for calculating power dissipation and the importance of RMS current in equating average power in AC and DC systems.