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Direct Current Circuits

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Electromotive Force

• The emf of a battery is the maximum possible voltage that the battery can provide between its terminals.

Circuit diagram of a source of emf (in

this case, a battery), of internal

resistance r, connected to an

external resistor of resistance R.

Graphical representation showing

how the electric potential changes

as the circuit in part (a) Is traversed

clockwise.

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Resistors in Series and Parallel

• For a series combination of two resistors, the currents are the same in both resistors because the amount of charge that passes through

R1 must also pass through R2 in the same time interval.

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• the equivalent resistance of a series connection of resistors is the numerical sum of the individual resistances and is always greater

than any individual resistance.

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Quick Quiz

• With the switch in the circuit of figure closed (left), there is no current in R2, because the current has an alternate zero-resistance path

through the switch. There is current in R1 and this current is

measured with the ammeter (a device for measuring current) at the right side of the circuit. If the switch is opened (right), there is current

in R2 . What happens to the reading on the ammeter when the switch

is opened?

– the reading goes up

– the reading goes down

– the reading does not change.

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• when resistors are connected in parallel, the potential differences across the resistors is the same.

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• the inverse of the equivalent resistance of two or more resistors connected in parallel is equal to the sum of the inverses of the

individual resistances. Furthermore, the equivalent resistance is

always less than the smallest resistance in the group.

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example

• A homeowner wishes to install 12-volt landscape lighting in his back yard. To save money, he purchases inexpensive 18-gauge cable, which has a relatively high resistance per unit length. This cable consists of two side-by-side wires separated by insulation, like the cord on an appliance. He runs a 200-foot length of this cable from the power supply to the farthest point at which he plans to position a light fixture. He attaches light fixtures across the two wires on the cable at 10-foot intervals, so the light fixtures are in parallel. Because of the cable’s resistance, the brightness of the bulbs in the light fixtures is not as desired. Which problem does the homeowner have?

– All of the bulbs glow equally less brightly than they would if lower resistance cable had been used.

– The brightness of the bulbs decreases as you move farther from the power supply.

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Kirchhoff’s Rules

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• Rules for determining the potential differences across a resistor and a battery. (The battery is assumed to have no internal resistance.)

Each circuit element is traversed from left to right.

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12

• A single-loop circuit contains two resistors and two batteries, as shown in figure. (Neglect the internal resistances of the batteries.)

Find the current in the circuit.

example

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RC Circuits

A capacitor in series with

a resistor, switch, and battery.

Circuit diagram representing

this system at time t < 0, before

the switch is closed.

Circuit diagram at time t > 0,

after the switch has been closed.

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• Plot of capacitor charge versus time for the circuit shown in figure before. After a time interval equal to one time constant τ has passed,

the charge is 63.2% of the maximum value Cε The charge

approaches its maximum value as t approaches infinity.

• Charge as a function of time for

a capacitor being charged

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• Plot of current versus time for the circuit shown in figure before. The current has its maximum value I0 =ε/R at t = 0 and decays to zero exponentially as t approaches infinity. After a time interval equal to one time constant τ has passed, the current is 36.8% of its initial value.

• Current as a function of time for

a capacitor being charged

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Discharging a Capacitor

• A charged capacitor connected to a resistor and a switch, which is open for t < 0. After the switch is closed at t = 0, a current that

decreases in magnitude with time is set up in the direction shown,

and the charge on the capacitor decreases exponentially with time.

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• Charge as a function of time for a discharging capacitor

• Current as a function of time for a discharging capacitor

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Electrical Meters

• The galvanometer is the main component in analog meters for

measuring current and voltage.

• The principal components of a

D’Arsonval galvanometer. When the coil

situated in a magnetic field carries a current, the magnetic torque causes the

coil to twist. The angle through which the coil rotates is proportional to the

current in the coil because of the

counteracting torque of the spring.

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The Ammeter

• Ideally, an ammeter should have zero resistance so that the current being measured is not altered.

• Current can be measured with an ammeter connected in series withthe elements in which the measurement of a current is desired.

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The Voltmeter

• An ideal voltmeter has infinite resistance so that no current exists in it.

• The potential difference across a resistor can be measured with a

voltmeter connected in parallel with the resistor

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Household Wiring and Electrical Safety

• Wiring diagram for a household circuit. The

resistances represent

appliances or other electrical devices that operate with an

applied voltage of 120 V.

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A diagram of the circuit for an electric drill with only two connecting wires.

The normal current path is from the live wire through the motor connections

and back to ground through the neutral wire. In the situation shown, the live wire has

come into contact with the drill case. As a result, the person holding the drill acts as a

current path to ground and receives an electric shock.

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This shock can be avoided by connecting the drill case to ground

through a third ground wire. In this situation, the drill case remains

at ground potential and no current exists in the person.

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quiz

• Under steady-state conditions, find the unknown currents I1, I2, and I3 in

the multiloop circuit shown in figure

(873)