A cylindrical shape resistance is connected to a battery with emg 5V. The resistance per unit length varies as `rho(x) = rho_(0) ((x)/(L))^(alpha)`, where `rho_(0) and alpha` are constant and x is the distance from one end of the resistor. The magnitude or product `rho_(0)L " is " 10 Omega`, where L is the length of the resistor. If the thermal power generated by the resistor is 20W, then the value of `alpha` is

Options

A. 3
B. 5
C. 7
D. 9

Answer

C

Explanation

Resistance of cylinder per unit length is
`(dR)/(dx) = rho ((x)/(L))^(alpha) rArr dR = rho ((x)/(L))^(alpha) .dx`
So, total resistance of cylinder,
`R = underset(0)overset(L)int dR = underset(0)overset(L)int (rho)/(L^(alpha)) .x^(alpha) dx = (rho)/(L^(alpha)) .((x^(alpha + 1))/(alpha + 1))_(0)^(L) = (rhoL^(alpha + 1))/(L^(alpha) (alpha + 1)) = (rhoL)/(alpha + 1)`
Power generated,
`P = (V^(2))/(R) = 20 rArr (25)/(((rho L)/(alpha + 1))) = 20`
`rArr (rho L)/(alpha + 1) = (5)/(4) rArr (10)/(alpha + 1) = (5)/(4) rArr alpha = 7`