Advertisements
Advertisements
Question
Two cells of emf E1, E2 and internal resistance r1 and r2 respectively are connected in parallel as shown in the figure.
Deduce the expressions for
(1) the equivalent e.m.f of the combination
(2) the equivalent resistance of the combination, and
(3) the potential difference between the point A and B.
Advertisements
Solution
Let I1 and I2 be the current in two cells with emf E1 and E2 and internal resistance r1 and r2
So I = I1 + I2
Now let V be the potential difference between the points A and B. Since the first cell is connected between the points A and B.
V = potential difference across first cell
V = E1 − I1r1
`or I_1 =(E_1-V)/r_1`
Now, the second cell is also connected between the points A and B. So,
`or I_2 =(E_2-V)/r_2`
Thus, substituting for I1 and I2
`I =(E_1-V)/r_1 + (E_2-V)/r_2`
`or, I (E_1 /r_1 + E_2 /r_2)-V(1 /r_1 +1 /r_2)`
`V =((E_1r_2 + E_2r_1)/(r_1+r_2)) -I((r_1r_2)/(r_1 +r_2)) ..... (1)`
If E is effective e.m.f and r, the effective internal resistance of the parallel combination of the two cells then V = E − Ir …..(2)
Comparing (1) and (2)
`(1) E= (E_1r_2 + E_2r_1)/(r_1+r_2)`
This is equivalent e.m.f of the combination
`(2)r =(r_1r_2)/(r_1 +r_2)`
This is equivalent resistance of the combination.
(3) The potential difference between the point A and B is
V = E − Ir
RELATED QUESTIONS
Two identical cells of emf 1.5 V each joined in parallel, supply energy to an external circuit consisting of two resistances of 7 Ω each joined in parallel. A very high resistance voltmeter reads the terminal voltage of cells to be 1.4 V. Calculate the internal resistance of each cell.
A battery of emf 10 V and internal resistance 3 Ω is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? What is the terminal voltage of the battery when the circuit is closed?
In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35.0 cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 63.0 cm, what is the emf of the second cell?
A secondary cell after long use has an emf of 1.9 V and a large internal resistance of 380 Ω. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car?
Nichrome and copper wires of same length and same radius are connected in series. Current I is passed through them. Which wire gets heated up more? Justify your answer.
Can the potential difference across a battery be greater than its emf?
Consider N = n1n2 identical cells, each of emf ε and internal resistance r. Suppose n1 cells are joined in series to form a line and n2 such lines are connected in parallel.
The combination drives a current in an external resistance R. (a) Find the current in the external resistance. (b) Assuming that n1 and n2 can be continuously varied, find the relation between n1, n2, R and r for which the current in R is maximum.
Three cells, each of emf E but internal resistances 2r, 3r and 6r are connected in parallel across a resistor R.
Obtain expressions for (i) current flowing in the circuit, and (ii) the terminal potential differences across the equivalent cell.
Study the two circuits shown in the figure below. The cells in the two circuits are identical to each other. The resistance of the load resistor R is the same in both circuits.
If the same current flows through the resistor R in both circuits, calculate the internal resistance of each cell in terms of the resistance of resistor R. Show your calculations.
