Electricity and Magnetism

Direct Current Circuits

A DC circuit consists of connected resistors and sources of emf (electromotive force) such as batteries. A lightbulb (a resistor) connected to a battery is a simple example of a DC circuit.

The question we ask is how to find the electric current in every branch of such a circuit.

We already know what the current is if we have a single resistor connected to a battery:

I = V/R

The most general cases need to addressed using Kirchhoff's rules to be discussed later. Before that, we want to learn how to find the equivalent resistance of resistors connected in series and in parallel. The concept of equivalent resistance simplifies the understanding of a circuit, and in many cases the simplification is already sufficient to allow us to find the currents without resorting to Kirchhoff's rules explicitly.

Combinations of resistors

Think of any combination of resistors as a black box. If we apply a voltage to it, there will a current (if everything is nicely connected). The ratio of the two is the equivalent resistance. We can replace this black box, however it is made up of, by a single resistor with the equivalent resistance and obtain the same current. Nothing has changed as far as other parts of the circuit is concerned.

Resistors in series

Req = R1 + R2 + R3 + ...


V = V1 + V2 + V3 + ...

and I is the same for all resistors.

Resistors in parallel

1/Req = 1/R1 + 1/R2 + 1/R3 + ...


I = I1 + I2 + I3 + ...

and V is the same for all resistors.

Specifically for two resistors

Req = R1R2/(R1+R2)

Note that

Req = (1/2)R

if R1 = R2 = R.

Compare the rules for equivalent resistance to those for equivalent capacitance and note the differences.

Example 18.3 Equivalent resistance

Solution of Example 18.4 using equivalent resistance

Kirchhoff's rules and complex DC circuits

For more complex DC circuits, we can find the current in every branch by applying Kirchhoff's rules:

The junction rule is a result of the conservation of charge. What goes in must come out since there can be no accumulation of charge at the junctions.

The loop rule is a result of the very definition of electric potential.

Example 18.4 Applying Kirchhoff's rules

Solution using Kirchhoff's rules

Example 18.5 Another application of Kirchhoff's rule

RC circuits

Examples of transient current

Charging of a capacitor

The circuit is shown in Figure 18.11.

The charge on the capacitor starts from zero and rises according to

q = Q( 1-e-t/t )

where e = 2.718 ... is the base of the natural logarithms; Q = CE is the maximum amount of charge we can put on the capacitor (with E); and

t = RC

is called the time constant of RC circuit. It is the amount of time it takes for the charge on the capacitor to reach 0.632Q, i.e., 63.2% of its maximum value Q.

The voltage as a function of time is given by

V(t) = q(t)/C = (Q/C)( 1-e-t/t )

Whatever we have said about charge q is equally applicable to voltage V since differ only by a constant of multiplication.

Discharging of a capacitor

The circuit is shown in Figure 18.22.

The charge on the capacitor decreases according to

q = Qe-t/t

After time t, the charge on the capacitor drops to (1/e)Q = 0.368Q (36.8% of its initial value). In other words, 63.2% of charge is lost after one time constant.

The voltage as a function of time is given by

V(t) = q(t)/C = (Q/C)e-t/t

Example 18.6 Charging a capacitor in an RC circuit

Example 18.7 Discharging a capacitor in a RC circuit

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Copyright 1997 by Bo Gao. All rights reserved.