Question

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In an experiment designed to measure the magnitude of a uniform magnetic field, electrons were accelerated from rest through a potential difference of 350V and then enter a magnetic field that is perpendicular to the velocity vector of the electrons. The electron travel along a curved path because of magnetic force exerted on them, and the radius of the path is measured to be 7.5 cm.

1. What will be the magnitude of the magnetic field?
2. What will be the angular speed of the electron?
3. What changes might occur to angular speed of electrons If there is sudden voltage surge causes the acceleration voltage to increase to 500v. assume magnetic field remains same.
4. What might happen if this experiment is conducted in non-uniform field. Sketch the motion of electron

Answer 1

-> potential difference, $\Delta$ V = 350 V

-> radius of path, r = 7.5 cm = 0.075 m

-> electron mass. m = 9.11*10-11 kg

-> charge of electron, q = -1.6*10-19 C

To find the magnitude of the magnetic field, we use the conservation of energy,

initial energy = final energy

(K + U)i = (K+U)f

(K + U)i - (K+U)f = 0

(Ki - Kf) + (Ui-Uf) = 0

(1/2*m*Vi2) -(1/2*m*Vf2) + (q*$\Delta$V ) = 0

(1/2*m*Vi2) -0 + (q*$\Delta$V ) = 0

0.5 * 9.11 *10-11 * Vi2+  (-1.6*10-19 )*350 = 0

=> Vi = 1.11*107 m/s

-> magnetic field, B = (m*Vi) / (q*r)

=> B = (9.11*10-11 * 1.11*107) / (1.6*10-19 * 0.075)

=> B = 8.4 * 10-4 T

-> The angular speed of the electron

w=Vir

$\omega =1.11*10^{7}/ 0.075 = 1.5*10^8\ rad/s$

-> If there is sudden voltage surge causing the acceleration voltage to increase to 500v,  will result in a speed increase of the electron which leads to the increase of the radius of the path taken by the electron. Hence, as both v and r increases, the angular speed which is (v/r) will neutralise and no change occurs to the angular speed.

-> If the experiment is conducted in a non uniform magnetic field, the motion of the electrons will be complex. the path taken by the electron can be a spiral and completely depends on the shape of the magnetic field.