TOPICS: Electric current, Ohm's law, resistance, combination of resistors, electrical power, work done, Kirchhoff's laws, galvanometer, ammeter, voltmeter, Wheatstone bridge, electromotive force and internal resistance.

Electric Current

An electric current exists between two points whenever one or more charges are transferred from one point to another. Switching an Electric bulb on, that lights up the room is the most familiar example of electric current. The tungsten filament of the bulb is heated up by passing current that results in light energy. The moving charges in the current are assumed to move with an average speed, called drift velocity. If an amount of charge q moves through a conductor in time t, then the current I is defined as,

I = dq/dt ................................................. (1)

The unit of current is ampere (A).

Ohm's law

The Ohm's law describes the simple relation between the current and potential in a conductor. The law may be stated as,

V = IR ................................................... (2)

where, V is the potential and R the resistance between two points in the conductor. The unit of resistance is Ohm.

The resistance R in a conductor is directly proportional to its length L and inversely proportional to the area of cross section A. That is,

R = rL/A ................................................. (3)

where r is called resistivity of the conductor. The inverse of resistivity is called conductivity. Unit of conductivity is inverse of Ohm that is "Mho". Below is a table of resistivity.

r = r_o[ 1 + a(T - T_o)] ............................... (4)

where a is called the temperature coefficient of resistivity.

R_eq = R₁ + R₂ + ................ +R_N ....................... (5)

When two or more resistors are in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances.

 

So for resistors in parallel,

1/Req = 1/R₁ + 1/R₂ + .............. + 1/R_N ........... (6)

When charges move in a conductor under the influence of electromotive force of a battery, work is done or power is delivered.

Small amount of Work done by the dq amount of charges is,

dW = V(dq)

where dt is the time for the charge to pass through. So. power

P = dW/(dt) = V(dq/dt) = VI

The electric power P delivered in a circuit is,

P = IV = I²R = V²/R using Ohm's law.................................... (7)

This power P is the same as heat loss per unit time in the resistor R of the circuit. Also, if q amount of charges move through a potential difference V, then the electrical work done is given by

W = qV ................................................. (8)

If positive charges move through a potential rise, the amount of work done is positive.

KIRCHHOFF'S RULES

The following two rules are known as Kirchhoff's laws.

1. Junction or Point rule: Sum of all currents entering a junction must equal sum of currents leaving the junction.

2. Loop or Circuit rule: For a closed loop in a circuit, the algebraic sum of all potential changes encountered while completing a cycle around the loop must be zero. In this Loop rue, we may consider a rise in the potential to be positive and a drop in the potential to be negative.

ELECTRIC METERS

The simple electrical instruments that measure currents and voltages, use a principal component called a Galvanometer. It consists of a resistor Rg that allows a small current to pass through it. The galvanometer current Ig passing through it, is calibrated to produce a readout on a scale. The readouts can be made digital or analog that uses a needle.

While building a meter, A Galvanometer is used together with a Shunt resistor R_s either in series or in parallel.

Ammeters: An ammeter measures current in a circuit. An Ammeter is constructed using a Galvanometer in parallel to the Shunt resistor. The Shunt resistor must be much smaller than Galvanometer resistor in it. This is done to bypass most of the current through the Shunt resistor. An ammeter must be connected in series with the circuit element.

In the arrangement for Ammeter, the voltage across the Shunt and Galvanometer is the same. Consequently,

I_gR_g = I_sR_s

Solving we get,

R_s = (I_g/I_s)R_g

Also, total current for full scale deflection is,

I = I_g + I_s

Voltmeters: To measure voltages in a circuit, a Voltmeter is constructed using a Galvanometer in series with the Shunt resistor. The Shunt resistor must be much greater than the Galvanometer resistor in it. This is done to pass almost all the current through the circuit. A Voltmeter must be connected in parallel to the circuit element under consideration.

In the arrangement for Voltmeter, the Shunt resistor and Galvanometer resistor are connected in series. Consequently, the current is the same through the Shunt and Galvanometer. As a result,

I_s = I_g

And total voltage V for full scale deflection is,

V = I_g(R_s + R_g)

That is, R_s = (V/I_g) - R_g

Electromotive Force

When a battery or a source of power delivers a current in the circuit, the terminal voltage of the battery is always less than the open circuit voltage of the source called the Electromotive force (emf). This is due to the Internal resistance present in all power sources. We may state,

Terminal voltage = emf - Ir

where r is the internal resistance of the source and I is the current delivered to the circuit.

Wheatstone Bridge:

Unknown resistance can be measured by a well known circuit called Wheatstone Bridge. The circuit consists of 3 known resistors R₁, R₂, R₃ and the unknown resistor R_x in the four arms of the circuit. A galvanometer is used to detect the current in the circuit as shown in the diagram below.

The ratio of the resistors R₁and R₂ are fixed while the value of the resistor R₃ is adjusted to obtain a zero current through the galvanometer. The Bridge is then balanced and the unknown resistance under the balanced Bridge condition, is given by,

R₂/R₁ = R_x/R₃

i.e., R_x = (R₂/R₁)R₃

EXERCISES

1. How many electrons are flowing every second through a light bulb if a steady current of 0.5 A passes through it.

2. A uniform current of 1.0 A passes through a section of a wire. (a) What is the magnitude of charge passing through the wire in 10 seconds? (b) What is the number of electrons passing through the wire in 10 seconds?

3. A steady current of 0.25 A passes through a cylindrical wire of radius 0.01 m. The current is uniformly distributed over the area of cross section. How many electrons per unit area are passing through the wire in 10 seconds?

4. An electron gun in a TV set bombards the screen with a current of 5x10^-6 A.

(a) How many electrons strike the the TV screen every second?

(b) How much charge falls on the TV screen in 30 seconds?

5. Find the potential difference between the two ends of a wire of resistance 15 Ohm if 5.0 C of charges move through it in 10 seconds.

6. An electric company runs two 5000 m long wires from the main point to a small factory for an estimated load current of 100A. Calculate the drop in the voltage suffered at the factory if the wire resistance is 0.0005 Ohm per 100 m.

7. An electric motor uses 3.0 A current when connected to a 115 V source. What is the power input of the motor?

8. An electric light bulb is rated 100 Watts. How much current will it use if connected to a 110 V source?

9. A 75 W bulb connected to a 115 V source outlet runs for 10 hours.

(a) What is the work done by it?

(b) What is the amount of charge passing through the bulb in 10 hours?

10. An electric heater of resistance 50 W uses 5.0 A of current when operates under steady condition. Compute its power if all the heat generated is due to the resistance.

11. A refrigerator uses 1.5 A of current at 115 V. How much of energy will it use in 30 days?

12. A television set uses 0.5 A when connected to a 110 V source. How much power will it consume if turned on?

13. A 110 V source supplies 0.5 A of current to a toaster. (a) What is its resistance? (b) How much power does it use? (c) What would be the cost of running it for 15 minutes every day for 30 days. Cost of 1 KWH is $0.15.

14. What is the potential drop across a 25 Ohm resistor if the current through it is 0.75 A?

15. A current of 3.0 A passes through point A to D in the following circuit, find the potential difference between the points,

(a) (C and D), (A and B), (A and C) and (A and D).

(b) Which point among A, B and C is at the highest potential?

16. Find the current and voltage across the 5.0 Ohm resistor in the following circuit when the switch S is turned on.

17. The resistance of a cylindrical wire is 25.0 Ohm. What will be the resistance of the wire of same material if the length is doubled but the diameter is kept cconstant.

18. The resistance of a certain wire is 10.0 Ohm. Find the resistance of the wire made out of same material if its diameter is doubled and the length is kept constant.

19. The length and diameter of a cylindrical wire is doubled. What would be the ratio of new and original resistance?

20. The resistivity of carbon is 3.5x10^-5 W.m. What should be the length of the carbon wire of 0.005 m in diameter if its total resistance is 3.5 W?

21. What is the equivalent resistance of 7 W, 9 W and 12 W resistors if connected in series?

22. The three resistors 25 W, 50 W and 75 W are connected in parallel. What is the resultant resistance of the three?

23. The electric circuit in a house connected to a source of 110 V outlet, has 4 bulbs in series. The power rating of the bulbs are 25 W, 50 W, 75 W and 100 W. What is the equivalent resistance of the 4 bulbs and the current in the circuit?

24. Find the equivalent resistance of the following circuit.

25. Find the current and the equivalent resistance in the following circuit when the switch S is closed.

26. (a) Find the equivalent resistance and the total current supplied by the cell in the following circuit, when switch is closed.

(b) Find also the current supplied by the cell in the 5.0 W resistors in the two branches..

27. A 6.0 Volt battery supplies 0.25 A of current when connected to a circuit. Find the emf of the battery if the internal resistance of the battery is 1.5 Ohm.

28. The internal resistance of a battery with emf of 3.0 V is 0.5 Ohm. Find the current and the terminal voltage of this battery if connected to a 100 ohm resistor.

29. A galvanomater produces a full scale deflection when 10.0 mA of current passes through it. It is to be used for constructing an Ammeter with full scale deflection of 10.0 A. If the galvanometer resistance is 5.0 Ohm, find (a) the shunt resistance needed, and (b) the total resistance of the Ammeter

30. A galvanometer has an internal resistance of 10.0 Ohm and uses 15.0 mA of current to give full scale deflection. It is used to build a Voltmeter that gives full scale deflection of 100 V. Find the voltage drop across the galvanometer and the shunt resistor needed to achieve it.

31. Find the three currents in the three branches of the following circuit.

32. A Wheatstone Bridge in the circuit below is balanced when when R₁= 15 Ohm, R₂=25 Ohm and R₃=40 Ohm. Find the unknown resistance.

33. In the Wheatstone Bridge Circuit diagram of problem #32, R₁=1000 Ohm and R₃=2.5R₂. Find the unknown resistance.

KEYS FOR PROBLEMS

Current 1,2,3,4,

Ohm's law 5,6,14,15

Power 7,8,9, 10,11,12,13,23

Kirchhoff's rules 16,25,26,31

Resistance 17,18,19,20,21,22,24,

Electromotive force 27,28

Electric meters 29,30

Wheatstone Bridge 32,33

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PICTURE SYMBOLS FOR ELECTRIC CIRCUITS:

Battery

Resistor

Capacitor

Inductor

RETURN TO GENERAL PHYSICS LECTURE NOTES

UPDATED 9/7/99