Voltage and Current in AC Circuits
Learning Goal: To understand the relationship between AC voltage and current in resistors, inductors, and capacitors, especially the phase shift between the voltage and the current.
In this problem, we consider the behavior of resistors, inductors, and capacitors driven individually by a sinusoidally alternating voltage source, for which the voltage is given as a function of time by . The main challenge is to apply your knowledge of the basic properties of resistors, inductors, and capacitors to these "single-element" AC circuits to find the current through each. The key is to understand the phase difference, also known as the phase angle, between the voltage and the current. It is important to take into account the sign of the current, which will be called positive when it flows clockwise from the b terminal (which has positive voltage relative to the a terminal) to the a terminal (see figure). The sign is critical in the analysis of circuits containing combinations of resistors, capacitors, and inductors.

Part A
First, let us consider a resistor with resistance connected to an AC source (diagram 1). If the AC source provides a voltage , what is the current through the resistor as a function of time?
Hint A.1 Ohm's law Ohm's law is still true at any moment in time.
 =
Note that the voltage and the current are in phase; that is, in the expressions for and , the arguments of the cosine functions are the same at any moment of time. This will not be the case for the capacitor and inductor.
Part B
Now consider an inductor with inductance in an AC circuit (diagram 2). Assuming that the current in the inductor varies as , find the voltage that must be driving the inductor.
Part B.1
Kirchhoff's loop rule
Apply Kirchhoff's loop rule to this circuit, going clockwise around the circuit (the same direction as the current arrow). Write down the sum of the voltage drops across each of the circuit elements.
Use for the voltage from the source, and describe the voltage drop across the inductor in terms of and
Part B.2 The derivative of Part not displayed
Hint B.3 The phase relationship between sine and cosine Hint not displayed
 =
Graphs of and are shown below. As you can see, for an inductor, the voltage leads (i.e., reaches its maximum before) the current by ; in other words, the current lags the voltage by . This can be conceptually understood by thinking of inductance as giving the current inertia: The voltage "tries" to push current through the inductor, but some sort of inertia resists the change in current. This is another manifestation of Lenz's law. The difference is called the phase angle.

Part C
Again consider an inductor with inductance connected to an AC source. If the AC source provides a voltage , what is the current through the inductor as a function of time?
Hint C.1 Using Part B You can obtain the answer almost immediately if you consider the results of Part B: The amplitude of the voltage ( ) is ; the frequency is the same as in Part B, and the phase difference is . Do you remember what leads and what lags?
 =
For the amplitudes (magnitudes) of voltage and current, one can write (for the resistor) and (for the inductor). If one compares these expressions, it should not come as a surprise that the quantity , measured in ohms, is called inductive reactance; it is denoted by (sometimes ). It is called reactance rather than resistance to emphasize that there is no dissipation of energy. Using this notation, we can write (for a resistor) and (for an inductor). Also, notice that the current is in phase with voltage when a resistor is connected to an AC source; in the case of an inductor, the current lags the voltage by . What will happen if we replace the inductor with a capacitor? We will soon see.
Part D
Consider the potentials of points a and b on the inductor in diagram 2. If the voltage at point b is greater than that at point a, which of the following statements is true?
Hint D.1 How to approach the problem Try drawing graphs of the current through the inductor and voltage across the inductor as functions of time.
 The current must be positive (clockwise). The current must be directed counterclockwise. The derivative of the current must be negative. The derivative of the current must be positive.
It may help to think of the current as having inertia and the voltage as exerting a force that overcomes this inertia. This viewpoint also explains the lag of the current relative to the voltage.
Part E
Assume that at time , the current in the inductor is at a maximum; at that time, the current flows from point b to point a. At time , which of the following statements is true?
Hint E.1 How to approach the problem Try drawing graphs of the current through the inductor and voltage across the inductor as functions of time.
 The voltage across the inductor must be zero and increasing. The voltage across the inductor must be zero and decreasing. The voltage across the inductor must be positive and momentarily constant. The voltage across the inductor must be negative and momentarily constant.
Part F
Now consider a capacitor with capacitance connected to an AC source (diagram 3). If the AC source provides a voltage , what is the current through the capacitor as a function of time?
Hint F.1 The relationship between charge and voltage for a capacitor Recall that by definition of capacitance, the relationship among charge, capacitance, and voltage is true at all times. The convention here is that the voltage is that on the plate with charge relative to the other plate (which has charge ).
Hint F.2 The relationship between charge and current No charged particles "flow" through the capacitor. Instead, the current deposits or drains charge from the plates. If the voltage is the voltage on terminal b (appropriate for Kirchhoff's law in this circuit), then if the current flow is positive (clockwise here as usual), the charge will increase. Hence , where is the charge on the capacitor plate b.
Hint F.3 Mathematical details To obtain the current, obtain from the voltage and then obtain the current from the time derivative of . Finally, rewrite the expression you obtain in terms of the cosine (instead of sine) function.
Express your answer in terms of , , and . Use the cosine function with a phase shift, not the sine function, in your answer.
 =
For the amplitude values of voltage and current, one can write . If one compares this expression with a similar one for the resistor, it should come as no surprise that the quantity , measured in ohms, is called capacitive reactance; it is denoted by (sometimes ). It is called reactance rather than resistance to emphasize that there is no dissipation of energy. Using this notation, we can write , and voltage lags current by radians (or 90 degrees). The notation is analogous to for a resistor, where voltage and current are in phase, and for an inductor, where voltage leads current by radians (or 90 degrees). We see, then, that in a capacitor, the voltage lags the current by , while in the case of an inductor, the current lags the voltage by the same quantity . In a capacitor, where voltage lags the current, you may think of the current as driving the change in the voltage.
Part G
Consider the capacitor in diagram 3. Which of the following statements is true at the moment the alternating voltage across the capacitor is zero?
Hint G.1 How to approach the problem Try drawing graphs of the (displacement) current through the capacitor and voltage across the capacitor as functions of time.
Hint G.2 Graphs of and
 The current must be directed clockwise. The current must be directed counterclockwise. The current must be at a maximum. The current must be zero.
Part H
Consider the capacitor in diagram 3. Which of the following statements is true at the moment the charge of the capacitor is at a maximum?
Hint H.1 Hint not displayed
Hint H.2 Hint not displayed
 The current must be directed clockwise. The current must be directed counterclockwise. The current must be at a maximum. The current must be zero.
Part I
Consider the capacitor in diagram 3. Which of the following statements is true if the voltage at point b is greater than that at point a?
Hint I.1 How to approach the problem Hint not displayed
Hint I.2 Hint not displayed
 The current must be directed clockwise. The current must be directed counterclockwise. The current may be directed either clockwise or counterclockwise.
Part J
Consider a circuit in which a capacitor and an inductor are connected in parallel to an AC source. Which of the following statements about the magnitude of the current through the voltage source is true?
Hint J.1 Driven AC parallel circuits The voltage across each element is the same at every moment in time. However, the magnitudes of the currents in an AC circuit cannot be added without consideration of the phase angle between the currents.