Physics 2000 JAMB Past Questions - Question 40: a cell of internal resistance 1Q supplies current to an external resistor of 3n. The efficiency of the cell is

Correct Answer

A

Explanation

Efficiency =R/1(R+r) *100=3/1+3 *100 =75%The efficiency of a cell, in the context of electrical circuits, is defined as the ratio of the useful power output to the total power input. In this case, we can calculate the efficiency of the cell supplying current to an external resistor.To calculate the efficiency of a cell, we need to know the power output and power input of the cell.

The power output (P_out) is given by the product of the current (I) and the voltage across the external resistor (V_R):

P_out = I * V_R

The power input (P_in) is given by the product of the current (I) and the total voltage across the cell (V_cell):

P_in = I * V_cell

The efficiency (η) of the cell is then given by the ratio of the power output to the power input:

η = P_out / P_in

In this case, the internal resistance of the cell is given as 1Ω, and the external resistor is 3Ω (n is not a recognized unit of resistance, so I assume it is a typo). Let's assume the voltage across the external resistor (V_R) is V.

The current (I) flowing through the circuit can be calculated using Ohm's Law:

I = V / (R + r)

where R is the resistance of the external resistor and r is the internal resistance of the cell.

Substituting the given values, we have:

I = V / (3Ω + 1Ω)
I = V / 4Ω

Now, we can calculate the power output (P_out) and power input (P_in):

P_out = I * V_R
P_out = (V / 4Ω) * V_R

P_in = I * V_cell
P_in = (V / 4Ω) * (V_R + V)

Finally, we can calculate the efficiency (η) using the formula:

η = P_out / P_in
η = [(V / 4Ω) * V_R] / [(V / 4Ω) * (V_R + V)]

Simplifying the equation, we get:

η = V_R / (V_R + V)

Therefore, the efficiency of the cell is given by the expression V_R / (V_R + V).

1. Calculate the current supplied by the cell:
  The current supplied by the cell (I_cell) can be calculated using Ohm's law:

  I_cell = V / (R_cell + R_external)

  Where:
  - V is the voltage of the cell.
  - R_cell is the internal resistance of the cell.
  - R_external is the resistance of the external resistor.

  In this case, V is the voltage of the cell, R_cell is 1 ohm, and R_external is 3 ohms (3n ohms is equivalent to 3 ohms). So:

  I_cell = V / (1 + 3)
  I_cell = V / 4

2. Calculate the power output (P_out):
  The power output is the power dissipated in the external resistor, which can be calculated using the formula:

  P_out = I_cell^2 * R_external

  Substituting the value of I_cell from step 1:

  P_out = (V / 4)^2 * 3
  P_out = (V^2 / 16) * 3

3. Calculate the power input (P_in):
  The power input is the power supplied by the cell, which can be calculated using the formula:

  P_in = I_cell^2 * (R_cell + R_external)

  Substituting the value of I_cell from step 1:

  P_in = (V / 4)^2 * (1 + 3)
  P_in = (V^2 / 16) * 4

4. Calculate the efficiency (η):
  Efficiency (η) is the ratio of the power output to the power input:

  η = P_out / P_in

  Substituting the expressions for P_out and P_in from steps 2 and 3:

  η = [(V^2 / 16) * 3] / [(V^2 / 16) * 4]

  Many terms cancel out:

  η = (3/4)

So, the efficiency of the cell is 3/4 or 75%.