Question

The aurora is caused when electrons and protons, moving in the earth's magnetic field of ≈5×10−5 T, collide with molecules of the atmosphere and cause them to glow. What is the radius of the circular orbit for (a) an electron with speed 1.0×106m/s and (b) a proton with speed 7.0×104m/s. What is re in meters and what is rp in meters?

Answer 1

Answer:

a) 0.114 meters is the radius of the circular orbit for an electron with given speed.

b) 14.6 meters is the radius of the circular orbit for a proton with given speed.

Explanation:

Force on the charge particle :

$F = qvB$

Since, the particle is revolving around in a circular path ;

$F'=\frac{mv^2}{r}$

particle in constantly moving in circular path under action of magnetic field:

F = F'

$qvB=\frac{mv^2}{r}$

$r=\frac{mv}{qB}$

Where :

r = radius of the circular path

m = mass of the particle

v = velocity of the particle

B = magnetic field

q = magnitude of  charge on the particle

a)  Velocity of the electron =  $v=1.0\times 10^6 m/s$

Mass of electron = m = $9.11\times 10^{-31} kg$

Charge on an electron , q= $1.602\times 10^{-19} C$

B = $5\times 10^{-5} T$

Radius of the circular orbit for an electron with given speed =$r_e$

$r_e=\frac{9.11\times 10^{-31} kg\times 1.0\times 10^6 m/s}{1.602\times 10^{-19} C\times 5\times 10^{-5} T}$

$r_e = 0.114 m$

0.114 meters is the radius of the circular orbit for an electron with given speed.

b)  Velocity of the proton =  $v=7.0\times 10^4 m/s$

Mass of electron = m = $1.67\times 10^{-27} kg$

Charge on an electron , q= $1.602\times 10^{-19} C$

B = $5\times 10^{-5} T$

Radius of the circular orbit for a proton with given speed =$r_p$

$r_e=\frac{1.67\times 10^{-27} kg\times 7.0\times 10^4 m/s}{1.602\times 10^{-19} C\times 5\times 10^{-5} T}$

$r_p = 14.6 m$

14.6 meters is the radius of the circular orbit for a proton with given speed.

Answer 2

The correct answer to this question is

You can use this equation.
f = m v^2 / r
f = q B v 

r = m v^2 / q B v = m v / q B = 

(a) an electron 9.109×10^-31 * 4.0×10^6 m/s / 1.602×10^-19 * 5.5*10^-5 T = 0.4135 m 
(b) a proton 1.673×10^-27 * 7.0×10^4 m/s ./ 1.602×10^-19 * 5.5×10^-5 T = 13.29 m