Question

A 1.5 m cylinder of radius 1.1 cm is made of a complicated mixture materials. Its resistivity depends on the distance x from the left and obeys the formula p(x)=a+bx^2, where a and b are constants. At the left, the resistivity is 2.25*10^-8 ohms*m, while at the right end its 8.5*10^-8 ohms*m
a)What is the resistance of the rod?
b)what is the electric field at its midpoint if it carries a 1.75-A current?
c) If we cut the rod into two 75-cm halves, what is the resistance of each half?

Answer 1

Answer:

a)$R = 171$μΩ

b)$E = 1.7 *10^{-4} V/m$

c)$R_{2} = 1.16 *10^{-4}$Ω

here * stand for multiplication

Explanation:

length of cylinder = 1.5 m

radius of cylinder  =  1.1 cm

resistivity depends on the distance x from the left

$p(x)=a+bx^2$ ............(i)

using equation

$R = \frac{pl}{a}$

let dR is the resistance of thickness dx

$dR =\frac{p(x)dx}{a}$

where p(x) is resistivity  l is length

a is area

$\int\limits^R_0 {dR} =\frac{1}{\pi r^2} \int\limits^L_0 {(a+bx^2)} \, dx$.........................(2)

after integration

$R = \frac{[aL+\frac{bL^3}{3}] }{\pi r^2}$  ...............(3)

it is given $p(0) = a = 2.25 * 10 ^{-8}$Ωm

$p(L) = a + b(L)^2$  = $8.5 * 10 ^{-8}$ Ωm

$8.5 * 10 ^{-8} = 2.25 * 10^{-8}+b(1.5)^2$

(here * stand for multiplication )

on solving we get

$b = 2.78* 10^{-8}$ Ωm

put each value of a  and b and r value in equation 3rd we get

$R = \frac{[aL+\frac{bL^3}{3}] }{\pi r^2}$

$R = 1.71 * 10^{-4}$Ω

$R = 171$μΩ

FOR (b)

for mid point  x = L/2

E = p(x)L

for x = L/2

$p(L/2) = a+b(L/2)^2$

for given current  I = 1.75 A

so electric field

 

$E = \frac{[a+b(L/2)^2]I }{\pi r^2}$

by substitute the values

we get;

$E = 1.7 *10^{-4} V/m$

(here * stand for multiplication )

c ).

75 cm means length will be half

 that is   x =  L/2

integrate  the second equation with upper limit  L/2  

Let resistance is $R_{1}$

so after integration we get

$R_{1} = \frac{[a(L/2) +(b/3)(L^3/8)]}{\pi r^2}$

substitute the value of a , b and L we get

$R_{1} = 5.47 * 10 ^{-5}$Ω

for second half resistance

$R_{2} = R- R_{1}$

$R_{2} = 1.7 *10^{-4} -5.47 *10^{-5}$

$R_{2} = 1.16 *10^{-4}$Ω

(here * stand for multiplication )