Question
A network of resistors is connected to a 16 V battery with internal resistance of 1 Ω as shown in fig.
(a) Compute the equivalent resistance of the network,
(b) Obtain the current in each resistor,
(c) Also obtain the voltage drops VAB, VBC and VCD.
Solution
Given, a network of resistors is connected to a 16 V battery with internal resistance 1Ω.
(a) The resistors are connected in parallel combination across AB.
Therefore, the equivalent resistance of two 4Ω resistors in parallel is
(a) The resistors are connected in parallel combination across AB.
Therefore, the equivalent resistance of two 4Ω resistors in parallel is
Requivalent = = 2 Ω.
Similarly, the resistors across CD are grouped in parallel combination.
Equivalent resistance of 12 Ω and 6 Ω resistors in parallel is,
Requivalent = = Ω = 4 Ω.
Now, 2 Ω,1 Ω and 4 Ω are in series combination.
So, total resistance is
Rseries = (2 + 1 + 4) Ω = 7 Ω .
(b) The current across the resistors can be calcualted using the formula,
Consider the resistors between A and B. It is a case of two equal resistors connected in parallel. So, current in each resistor is 1 A.
Current through 1 Ω is clearly 2 A.
Similarly, the resistors across CD are grouped in parallel combination.
Equivalent resistance of 12 Ω and 6 Ω resistors in parallel is,
Requivalent = = Ω = 4 Ω.
Now, 2 Ω,1 Ω and 4 Ω are in series combination.
So, total resistance is
Rseries = (2 + 1 + 4) Ω = 7 Ω .
(b) The current across the resistors can be calcualted using the formula,
Consider the resistors between A and B. It is a case of two equal resistors connected in parallel. So, current in each resistor is 1 A.
Current through 1 Ω is clearly 2 A.
Let us now consider resistors between C and D. It is a parallel combination of two resistances. Current would be divided in the inverse ratio of resistances.
If I1 is the current through 12 Ω and I2 is the current through 6 Ω, then,
So, current through 12 Ω resistor is
Similarly, current through 6 Ω resistor is
If I1 is the current through 12 Ω and I2 is the current through 6 Ω, then,
So, current through 12 Ω resistor is
Similarly, current through 6 Ω resistor is
(c) The voltage VAB between A and B is the product of total current between A and B and the equivalent resistance between A and B.
∴ VAB = 2 x 2 = 4V
Similarly,
Voltage drop across B and C, VBC = 2 x 1 = 2V
Voltage drop across C and D, VCD = 2 x 4 =8V
Note that the terminal voltage is 14V. The loss of 2V is due to internal resistance of battery.
