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Direct Current Circuits: 3-1 EMF

3-2 Resistance in series and parallel .

3-3 Rc circuit

3-4 Electrical instruments

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Example: (A) Find the current in the circuit.

starting at a, we see that a b represents a potential differenceof + Ɛ1

b c represents a potential difference of -IR1, c d represents a potential difference of - Ɛ2, andd a represents a potential difference of -IR2

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(B) What power is delivered to each resistor? What power is delivered by the 12-V battery?

The negative sign for I indicates that the direction of the

current is opposite the assumed direction.

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Problem-Solving Strategy – Kirchhoff’s RulesDraw the circuit diagram and assign

labels and symbols to all known and unknown quantities

Assign directions to the currents.Apply the junction rule to any junction in

the circuitApply the loop rule to as many loops as

are needed to solve for the unknownsSolve the equations simultaneously for

the unknown quantitiesCheck your answers3-12-2014 5FCI

Example:

A) Under steady-state conditions, find the unknown currents

I1, I2, and I3 in the multiloop circuit shown in Figurev

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(B) What is the charge on the capacitor?

We can apply Kirchhoff’s loop rule to loop bghab(or any other loop that contains the capacitor) to find thepotential difference ∆Vcap across the capacitor. We use this potential difference in the loop equation without reference to a sign convention because the charge on the capacitor depends only on the magnitude of the potential difference.Moving clockwise around this loop, we obtain

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Quiz 3:In using Kirchhoff’s rules, you generally

assign a separate unknown current to (a) each resistor in the circuit (b) each loop in

the circuit (c) each branch in the circuit (d) each battery in the circuit.

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Ans.(c) each branch in the circuit

A current is assigned to a given branch of a circuit.

There may be multiple resistors and batteries in a given branch.

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RC CircuitsA direct current circuit may contain

capacitors and resistors, the current will vary with time

When the circuit is completed, the capacitor starts to charge

The capacitor continues to charge until it reaches its maximum charge (Q = Cε)

Once the capacitor is fully charged, the current in the circuit is zero

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Charging Capacitor in an RC CircuitThe charge on the

capacitor varies with timeq = Q(1 – e-t/RC)The time constant, =RC

The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum

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The following dimensional analysis shows that τ has the units of time:

The energy output of the battery as the capacitor is fully

charged is Q Ɛ= Ɛ C 2.

After the capacitor is fully charged, the energy stored in

the capacitor is 1/2Q Ɛ = 1/2 CƐ 2, which is just half the

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Notes on Time Constant

In a circuit with a large time constant, the capacitor charges very slowly

The capacitor charges very quickly if there is a small time constant

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Discharging Capacitor in an RC CircuitWhen a charged capacitor is

placed in the circuit, it can be discharged q = Q e-t/RC

The charge decreases exponentially

At t = = RC, the charge decreases to 0.368 Qmax

In other words, in one time constant, the capacitor loses 63.2% of its initial charge

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5- Electric meter

Ammeters and Voltmeters

An ammeter is a device for measuring current, and a voltmeter measures voltages.

The current in the circuit must flow through the ammeter; therefore the ammeter should have as low a resistance as possible, for the least disturbance.

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Ammeters and Voltmeters

A voltmeter measures the potential drop between two points in a circuit. It therefore is connected in parallel; in order to minimize the effect on the circuit, it should have as large a resistance as possible.

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Summary

• Power in an electric circuit:

• If the material obeys Ohm’s law,

• Energy equivalent of one kilowatt-hour:

• Equivalent resistance for resistors in series:

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Summary

• Junction rule: All current that enters a junction must also leave it.

• Loop rule: The algebraic sum of all potential charges around a closed loop must be zero.

• Inverse of the equivalent resistance of resistors in series:

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Summary

• Equivalent capacitance of capacitors connected in parallel:

• Inverse of the equivalent capacitance of capacitors connected in series:

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Kirchhoff’s rules:

1- Junction rule. The sum of the currents entering any junction in an electric circuit must equal the sum of the currents leaving that junction:

2- Loop rule. The sum of the potential differences across all elements around any circuit loop must be zero:

The first rule is a statement of conservation of charge; the second is equivalent to a statement of conservation of energy.

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Note: When a resistor is traversed in the direction of the current, the potential difference ∆V across the resistor is -IR.

When a resistor is traversed in the direction opposite thecurrent, ∆V = +IR.

When a source of emf is traversed in the direction of the emf (negative terminal to positive terminal), the potential difference is +Ɛ .

When a source of emf is traversed opposite the emf (positive to negative), the potential difference is -Ɛ

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• Charging a capacitor:

• Discharging a capacitor:

where Q = C Ɛ is the maximum charge on the capacitor. The product RC is called the time constant Ƭ of the circuit.

where Q is the initial charge on the capacitor and I0 = Q /RC is the initial current in the circuit.

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Summary

• Ammeter: measures current. Is connected in series. Resistance should be as small as possible.

• Voltmeter: measures voltage. Is connected in parallel. Resistance should be as large as possible.

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Household CircuitsThe utility company

distributes electric power to individual houses with a pair of wires

Electrical devices in the house are connected in parallel with those wires

The potential difference between the wires is about 120V

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