Question

Two heaters A and B have power rating of 1 kW and 2 kW, respectively. Those two are first connected in series and then in parallel to a fixed power source. The ratio of power outputs for these two cases is ______

Options

• 1 : 2

• 2 : 3

• 1 : 1

• 2 : 9

Answer 1

Two heaters A and B have power rating of 1 kW and 2 kW, respectively. Those two are first connected in series and then in parallel to a fixed power source. The ratio of power outputs for these two cases is 2 : 9.

Explanation:

The resistance R of an appliance with power rating P at voltage V is given by:

R = `V^2/P`

Assuming both heaters have the same rated voltage:

Heater A (P_A = 1 kW)

R_A = `V^2/1000`

Heater B (PB = 2 kW)

R_B = `V^2/2000`

Let, R_B = R

Then, R_A = 2R

In a series connection, the total resistance is the sum of individual resistances:

`R_"series"` = R_A + RB

= 2 R + R

= 3 R

The total power output in series is:

`P_"series" = V^2/R_"series"`

= `V^2/(3 R)`

In a parallel connection, the total resistance is:

`R_"parallel" = (R_A * R_B)/(R_A + R_B)`

= `(2 R * R)/(2 R + R)`

= `(2 R^2)/(3 R)`

= `2/3 R`

The total power output in parallel is:

`P_"parallel" = V^2/R_"parallel"`

= `V^2/(2/3 R)`

= `(3 V^2)/(2 R)`

Alternatively, power in parallel is simply the sum of individual powers:

`P_"parallel"` = 1 kW + 2 kW

= 3 kW

The ratio of series power to parallel power is:

Ratio = `P_"series"/P_"parallel"`

= `(V^2/(3 R))/((3 V^2)/(2 R))`

= `V^2/(3 R) * (2 R)/(3 V^2)`

= `2/9`