Physics : Current Electricity: Solved Example Numerical problems with Answers, Solution and Explanation

1. The following graphs represent the current versus voltage and voltage versus current for the six conductors A,B,C,D,E and F. Which conductor has least resistance and which has maximum resistance?

Ans: Least: RF = 0.4 Ω, maximum RC = 2.5 Ω

2. Lightning is very good example of natural current. In typical lightning, there is 109 J energy transfer across the potential difference of 5 × 107 V during a time interval of 0.2 s.

Using this information, estimate the following quantities (a) total amount of charge transferred between cloud and ground (b) the current in the lightning bolt (c) the power delivered in 0.2 s.

Ans: charge = 20 C, I = 100 A, P = 5 GW

3. A copper wire of 10-6 m2 area of cross section, carries a current of 2 A. If the number of electrons per cubic meter is 8 × 1028, calculate the current density and average drift velocity.

Ans: J = 2 × 106 Am−2 ; vd= 15.6 × 10−5 ms−1

4. The resistance of a nichrome wire at 0º C is 10 Ω. If its temperature coefficient of resistance is 0.004/ºC, find its resistance at boiling point of water. Comment on the result.

Ans: RT= 14 Ω. As the temperature increases the resistance of the wire also increases.

5. The rod given in the figure is made up of two different materials.

Both have square cross sections of 3 mm side. The resistivity of the first material is 4 x 10-3 Ω.m and it is 25 cm long while second material has resistivity of 5 x 10-3 Ω.m and is of 70 cm long. What is the resistivity of rod between its ends?

Ans: 500 Ω

6. Three identical lamps each having a resistance R are connected to the battery of emf as shown in the figure.

Suddenly the switch S is closed. (a) Calculate the current in the circuit when S is open and closed (b) What happens to the intensities of the bulbs A,B and C. (c) Calculate the voltage across the three bulbs when S is open and closed (d) Calculate the power delivered to the circuit when S is opened and closed (e) Does the power delivered to the circuit decreases, increases or remain same?

Ans:

7. The current through an element is shown in the figure. Determine the total charge that pass through the element at a) t = 0 s, b) t = 2 s, c) t = 5s

Ans: At t= 0s,dq = 0 C, At t=2 s, dq = 10 C; At t=5 s, dq = 0 C

8. An electronics hobbyist is building a radio which requires 150 Ω in her circuit, but she has only 220 Ω, 79 Ω and 92 Ω resistors available. How can she connect the available resistors to get desired value of resistance?

Ans: Parallel combination of 220 Ω and 79 Ω in series with 92 Ω

9. A cell supplies a current of 0.9 A through a 2 Ω resistor and a current of 0.3 A through a 7 Ω resistor. Calculate the internal resistance of the cell.

Ans: 0.5 Ω

10. Calculate the currents in the following circuit.

Ans : I1 = 0.070 A, I2 = -0.010 A and I3 = 0.080 A

11. A potentiometer wire has a length of 4 m and resistance of 20 Ω. It is connected in series with resistance of 2980 Ω and a cell of emf 4 V. Calculate the potential along the wire.

Ans: Potential = 0.65 × 10-2 V m-1.

12. Determine the current flowing through the galvanometer (G) as shown in the figure.

13. Two cells each of 5V are connected in series across a 8 Ω resistor and three parallel resistors of 4 Ω, 6 Ω and 12 Ω. Draw a circuit diagram for the above arrangement. Calculate i) the current each resistor

Ans: The current at 4 Ω ,I = 2/4 = 0 .5A, the current at 6 Ω, I = 2/6 = 0.33A , the current at 12 Ω, I = 2/12 = 0 . 17 A

14. Four light bulbs P, Q, R, S are connected in a circuit of unknown arrangement. When each bulb is removed one at a time and replaced, the following behavior is observed.

Draw the circuit diagram for these bulbs.

Ans

15. In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35 cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 63 cm, what is the emf of the second cell?

Ans: emf of the second cell is 2.25 V

Compute the current in the wire if a charge of 120 C is flowing through a copper wire in 1 minute.

Solution

The current (rate of flow of charge) in the wire is

I = Q/t = 120/60 = 2A

If an electric field of magnitude 570 N C-1, is applied in the copper wire, find the acceleration experienced by the electron.

E = 570 N C-1, e = 1.6 × 10-19 C, m = 9.11 × 10-31 kg and a = ?

F = ma = eE

a = eE/*m* = 570×1 .6×10−19/9 .11×10-31

= 912 ×10−19 ×1031 / 9 .11

= 1.001 × 1014 m s-2

A copper wire of cross-sectional area 0.5 mm2 carries a current of 0.2 A. If the free electron density of copper is 8.4 × 1028 m-3 then compute the drift velocity of free electrons.

The relation between drift velocity of electrons and current in a wire of cross-sectional area A is

vd = I/ ne A

vd = 0.03 x 10-3 m s-1

Determine the number of electrons flowing per second through a conductor, when a current of 32 A flows through it.

I = 32 A , t = 1 s

Charge of an electron, e = 1.6 × 10-19 C

The number of electrons flowing per second, n =?

I = q/t = *ne*/*t*

n = It/e

n = 32×1 / 1 .6×10−19 C

n = 20 × 1019 = 2 × 1020 electrons

EXAMPLE 2.5

A potential difference across 24 Ω resistor is 12 V. What is the current through the resistor?

Solution

V = 12 V and R = 24 Ω

Current, I = ?

From Ohm’s law, I = V/R = 12/24 = 0 .5 A

EXAMPLE 2.6

The resistance of a wire is 20 Ω. What will be new resistance, if it is stretched uniformly 8 times its original length?

Solution

R1 = 20 Ω, R2= ?

Let the original length (l1) be *l*.

The new length, *l*2 = 8*l*1 (i.,e) l2 =8*l*

The original resistance, R = ρ [ *l*1 / A1]

The new resistance

R2 =

Though the wire is stretched, its volume is unchanged.

Initial volume = Final volume

A1*l*1= A2*l*2 , A1*l* = A28*l*

A1 / A2 = 8*l* /* l *= 8

By dividing equation R2 by equation R1, we get

Substituting the value of A1/A2, we get

R2 / R1 = 8 ×8 = 64 2

R2 = 64 × 20=1280 Ω

Hence, stretching the length of the wire has increased its resistance.

EXAMPLE 2.7

Consider a rectangular block of metal of height A, width B and length C as shown in the figure.

If a potential difference of V is applied between the two faces A and B of the block (figure (a)), the current IAB is observed. Find the current that flows if the same potential difference V is applied between the two faces B and C of the block (figure (b)). Give your answers in terms of IAB.

Solution

In the first case, the resistance of the block

EXAMPLE 2.8

Calculate the equivalent resistance for the circuit which is connected to 24 V battery and also find the potential difference across 4 Ω and 6 Ω resistors in the circuit.

Solution

Since the resistors are connected in series, the effective resistance in the circuit

= 4 Ω + 6 Ω = 10 Ω

The Current I in the circuit= V/ Req = 24/10 = 2 .4 A

Voltage across 4Ω resistor

V1 = IR1 = 2 . 4 A× 4 Ω = 9.6V

Voltage across 6 Ω resistor

V2 = IR1 = 2 . 4 A× 6 Ω =14 .4V

EXAMPLE 2.9

Calculate the equivalent resistance in the following circuit and also find the current I, I1 and I2 in the given circuit.

Solution

Since the resistances are connected in parallel, therefore, the equivalent resistance in the circuit is

The resistors are connected in parallel, the potential (voltage) across each resistor is the same.

The current I is the total of the currents in the two branches. Then,

I = I1 + I2= 6 A + 4 A = 10 A

EXAMPLE 2.10

When two resistances connected in series and parallel their equivalent resistances are 15 Ω and 56/15 Ω respectively. Find the individual resistances.

Solution

Rs = R1 + R2 = 15 Ω (1)

The above equation can be solved using factorisation.

R12-8 R1-7 R1+ 56 = 0

R1 (R1– 8) – 7 (R1– 8) = 0

(R1– 8) (R1– 7) = 0

If (R1= 8 Ω)

using in equation (1)

8 + R2 = 15

R2 = 15 – 8 = 7 Ω ,

R2 = 7 Ω i.e , (when R1 = 8 Ω ; R2 = 7 Ω)

If (R1= 7 Ω)

Substituting in equation (1)

7 + R2 = 15

R2 = 8 Ω , i.e , (when R1 = 8 Ω ; R2 = 7 Ω )

EXAMPLE 2.11

Calculate the equivalent resistance between A and B in the given circuit.

Solution

EXAMPLE 2.12

Five resistors are connected in the configuration as shown in the figure. Calculate the equivalent resistance between the points a and b.

Solution

Case (a)

To find the equivalent resistance between the points a and b, we assume that current is entering the junction a. Since all the resistances in the outside loop are the same (1Ω), the current in the branches ac and ad must be equal. So the electric potential at the point c and d is the same hence no current flows into 5 Ω resistance. It implies that the 5 Ω has no role in determining the equivalent resistance and it can be removed. So the circuit is simplified as shown in the figure.

The equivalent resistance of the circuit between a and b is Req = 1Ω

EXAMPLE 2.13

If the resistance of coil is 3 Ω at 20 0C and α = 0.004/0C then determine its resistance at 100 0C.

Solution

R0= 3 Ω, T = 100ºC, T0 = 20ºC

α = 0.004/ºC, RT = ?

RT= R0(1 + α(T-T0))

R100 = 3(1 + 0.004 × 80)

R100 = 3(1 + 0.32)

R100 = 3(1.32)

R100 = 3.96 Ω

EXAMPLE 2.14

Resistance of a material at 10ºC and 40ºC are 45 Ω and 85 Ω respectively. Find its temperature co-efficient of resistance.

Solution

T0 = 10ºC, T = 40ºC, R0= 45 Ω , R = 85 Ω

α = 1/R . ΔR /ΔT

α = 0.0296 per ºC

EXAMPLE 2.15

A battery of voltage V is connected to 30 W bulb and 60 W bulb as shown in the figure. (a) Identify brightest bulb (b) which bulb has greater resistance? (c) Suppose the two bulbs are connected in series, which bulb will glow brighter?

Solution

(a) The power delivered by the battery P = VI. Since the bulbs are connected in parallel, the voltage drop across each bulb is the same. If the voltage is kept fixed, then the power is directly proportional to current (P ∝ I). So 60 W bulb draws twice as much as current as 30 W and it will glow brighter then others.

(b) To calculate the resistance of the bulbs, we use the relation P = V2 / R . In both the bulbs, the voltage drop is the same, so the power is inversely proportional to the resistance or resistance is inversely proportional to the power (R∝1/P) . It implies that, the 30W has twice as much as resistance as 60 W bulb.

(c) When these two bulbs are connected in series, the current passing through each bulb is the same. It is equivalent to two resistors connected in series. The bulb which has higher resistance has higher voltage drop. So 30W bulb will glow brighter than 60W bulb. So the higher power rating does not always imply more brightness and it depends whether bulbs are connected in series or parallel.

EXAMPLE 2.16

Two electric bulbs marked 20 W – 220 V and 100 W – 220 V are connected in series to 440 V supply. Which bulb will be fused?

Solution

To check which bulb will be fused, the voltage drop across each bulb has to be calculated.

The resistance of a bulb,

Both the bulbs are connected in series. So the current which passes through both the bulbs are same. The current that passes through the circuit, I = V /Rtot.

Rtot = ( R1 + R2 )

Rtot = ( 484 +2420) Ω = 2904Ω

I = 440V/2904Ω ≈ 0.151A

The voltage drop across the 20W bulb is

V1 = IR1 = 440/2904 × 2420 ≈ 366.6 V

The voltage drop across the 100W bulb is

V2 = IR2 = 440/2904 × 484 ≈ 73.3 V

The 20 W bulb will be fused because its voltage rating is only 220 V and 366.6 V is dropped across it.

Determination of internal resistance: Solved Example Problems

EXAMPLE 2.17

A battery has an emf of 12 V and connected to a resistor of 3 Ω. The current in the circuit is 3.93 A. Calculate (a) terminal voltage and the internal resistance of the battery (b) power delivered by the battery and power delivered to the resistor

Solution

The given values I = 3.93 A, ξ = 12 V, R = 3 Ω

(a) The terminal voltage of the battery is equal to voltage drop across the resistor

V = IR = 3.93 × 3 = 11.79 V

The internal resistance of the battery,

r = |ξ –V / V| R = | 12 −11 .79 /11 .79 | × 3 = 0.05 Ω

The power delivered by the battery P = Iξ = 3.93 × 12 = 47.1 W

The power delivered to the resistor = I2 R = 46.3 W

The remaining power = (47.1 – 46.3) P = 0.772 W is delivered to the internal resistance and cannot be used to do useful work. (it is equal to I2 r).

EXAMPLE 2.18

From the given circuit,

Find

i) Equivalent emf of the combination

ii) Equivalent internal resistance

iii) Total current

iv) Potential difference across external resistance

v) Potential difference across each cell

Solution

i) Equivalent emf of the combination ξeq = nξ = 4 9 = 36 V

ii) Equivalent internal resistance req = nr = 4 × 0.1 = 0.4 Ω

iii) Total current I = nξ / R +nr

= [4 ×9] / 10 + ( 4 ×0.1)

= [4 ×9] / [10 +0 .4] = 36 /10.4

I = 3.46 A

iv) Potential difference across external resistance V = IR = 3.46 × 10 = 34.6 V. The remaining 1.4 V is dropped across the internal resistance of cells.

v) Potential difference across each cell V/*n* = 34.6/4 = 8 .65V

EXAMPLE 2.19

From the given circuit

Find

i) Equivalent emf

ii) Equivalent internal resistance

iii) Total current (I)

iv) Potential difference across each cell

v) Current from each cell

Solution

i) Equivalent emf ξeq = 5 V

ii) Equivalent internal resistance,

Req = r/n = 0 .5/4 = 0.125Ω

iii) total current,

I ≈ 0.5 A

iv) Potential difference across each cell V = IR = 0.5 × 10 = 5 V

v) Current from each cell, I ′ = I/n

I ′ = 0.5/4 = 0.125 A

EXAMPLE 2.20

From the given circuit find the value of I.

Solution

Applying Kirchoff’s rule to the point P in the circuit,

The arrows pointing towards P are positive and away from P are negative.

Therefore, 0.2A – 0.4A + 0.6A – 0.5A + 0.7A – I = 0

1.5A – 0.9A – I = 0

0.6A – I = 0

I = 0.6 A

EXAMPLE 2.21

The following figure shows a complex network of conductors which can be divided into two closed loops like ACE and ABC. Apply Kirchoff’s voltage rule.

*Solution*

Thus applying Kirchoff’s second law to the closed loop EACE

*I*1*R*1* *+* I*2*R*2* *+* I*3*R*3* *=* *ξ

and for the closed loop ABCA

*I*4*R*4* *+* I*5*R*5-*I*2*R*2= 0

EXAMPLE 2.22

Calculate the current that flows in the 1 Ω resistor in the following circuit.

*Solution*

We can denote the current that flows from 9V battery as I1 and it splits into I2 and I1 – I2 in the junction according Kirchoff’s current rule (KCR). It is shown below.

Now consider the loop EFCBE and apply KVR, we get

1I2 + 3I1 + 2I1 = 9

5I1 + I2 = 9 (1)

Applying KVR to the loop EADFE, we get

3 (I1 – I2 ) – 1I2 = 6

3I1 – 4I2 = 6 (2)

Solving equation (1) and (2), we get

I1 = 1.83 A and I2 = -0.13 A

It implies that the current in the 1 ohm resistor flows from F to E.

EXAMPLE 2.23

In a Wheatstone’s bridge P = 100 Ω, Q = 1000 Ω and R = 40 Ω. If the galvanometer shows zero deflection, determine the value of S.

*Solution*

EXAMPLE 2.24

What is the value of *x* when the Wheatstone’s network is balanced?

P = 500 Ω, Q = 800 Ω, R = *x* + 400, S = 1000 Ω

*Solution*

P/Q = R/S

x + 400 = 0.625 × 1000

x + 400 = 625

x = 625 – 400

x = 225 Ω

EXAMPLE 2.25

In a meter bridge with a standard resistance of 15 Ω in the right gap, the ratio of balancing length is 3:2. Find the value of the other resistance.

*Solution*

EXAMPLE 2.25

In a meter bridge, the value of resistance in the resistance box is 10 Ω. The balancing length is *l*1 = 55 cm. Find the value of unknown resistance.

*Solution*

Q = 10 Ω

EXAMPLE 2.27

Find the heat energy produced in a resistance of 10 Ω when 5 A current flows through it for 5 minutes.

Solution

R = 10 Ω, I = 5 A, t = 5 minutes = 5 × 60 s

H = I2 R t

= 52 × 10 × 5 × 60

=25 × 10 × 300

=25 × 3000

=75000 J (or) 75 kJ

EXAMPLE 2.28

An electric heater of resistance 10 Ω connected to 220 V power supply is immersed in the water of 1 kg. How long the electrical heater has to be switched on to increase its temperature from 30°C to 60°C. (The specific heat of water is s = 4200 J kg-1)

Solution

According to Joule’s heating law H = I2 Rt

The current passed through the electrical heater = 220V/10Ω = 22 A

The heat produced in one second by the electrical heater H = I2 R

The heat produced in one second H = (22)2 x 10 = 4840 J = 4.84 k J. In fact the power rating of this electrical heater is 4.84 k W.

The amount of energy to increase the temperature of 1kg water from 30°C to 60°C is

Q = ms ∆T (Refer XI physics vol 2, unit 8)

Here m = 1 kg,

s = 4200 J kg-1,

∆T = 30,

so Q = 1 × 4200 x 30 = 126 kJ

The time required to produce this heat energy t = Q/ I2R = 126 ×103 / 4840 ≈ 26 .03 s

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