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

a)
`R = 171`μΩ

b)
`E = 1.7 *10^(-4) V/m`

c)
`R_(2) = 1.16 *10^(-4)`Ω

here * stand for multiplication

Step-by-step 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 = (pl)/(a)`

let dR is the resistance of thickness dx

`dR =(p(x)dx)/(a)`

where p(x) is resistivity l is length

a is area

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

after integration

`R = ([aL+(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 = ([aL+(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 = ([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) = ([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 )