Question

Resistance x ohms, 50 ohms and 15 ohms are connected in parallel, the current through the resistor x is 2 amps and the total current taken from the supply mains is 15 amps. Find

a) the value of the unknown resistance x,

b) the p.d across the circuit,

c) the current in each resistance,

d) the total resistance of the current

Answer 1

a) If the three resistors are connected in parallel, then the total resistance R may be calculated as

$\dfrac1R = \dfrac1x + \dfrac{1}{15} + \dfrac{1}{50} , \;\; R = \dfrac{750x}{750+65x}$ .

The total potential difference across the circuit is $U=RI$ and the same potential difference is across every resistor. The p.d. across the first resistor is $U = xI_1 .$ Therefore we get an equation

$x\cdot 2 = \dfrac{750x}{750+65x}\cdot 15$ , $1500x + 130x^2 = 1250 x,\;\; 130x = 9750x, \;\; 130x = 9750, \;\; x = 75\,\mathrm{Ohm}.$

b) The total p.d. is equal to p.d. on the first resistor $U = xI_1 = 75\,\mathrm{Ohm}\cdot2\,\mathrm{A} = 150\,\mathrm{V}.$

c) The current in the second resistor is $I_2 = \dfrac{U}{R_2} = \dfrac{150\,\mathrm{V}}{50\,\mathrm{Ohm}} = 3\,\mathrm{A}\,,$ the current in the third resistor $I_3 = \dfrac{U}{R_2} = \dfrac{150\,\mathrm{V}}{15\,\mathrm{Ohm}} = 10\,\mathrm{A}\,.$

d) The total resistance is $R = \dfrac{750\cdot 75}{750+65\cdot 75} = 10\,\mathrm{Ohm}.$