A proton moves perpendicular to a uniform magnetic field B at a speed of 2.40 107 m/s and experiences an acceleration of 1.90 1013 m/s2 in the positive x direction when its velocity is in the positive z direction. Determine the magnitude and direction of the field.

Magnitude of the magnetic field = B = 0.00827 T = 8.27 m T

Direction = B is directed in the negative y-direction.

Explanation:

Given

Charge moving through the magnetic field is a proton, Hence,

Charge = (1.602 × 10⁻¹⁹) C

Mass of the proton = (1.673 × 10⁻²⁷) kg

Magnitude of velocity = v = (2.40 × 10⁷) m/s

Acceleration = (1.90 × 10¹³) m/s²

Angle between the proton's motion and the magnetic field = 90°

First of, we find the magnitude of the magnetic field.

F = qvB sin θ

where

F = magnitude of the force experienced by the proton = ma = (mass of a proton) × (acceleration of the proton)

F = ma = (1.673 × 10⁻²⁷ × 1.90 × 10¹³)

F = (3.179 × 10⁻¹⁴) N

(3.179 × 10⁻¹⁴) = (1.602 × 10⁻¹⁹ × 2.40 × 10⁷ × B × sin 90°)

B = 0.00827 T = 8.27 m T

To get the direction of the magnetic field, we can use the right hand rule, or the vector multiplication method.

The right hand rule expresses the directions of the velocity of proton, force on the proton and the magnetic field using the first 3 fingers on the right hand (the thumb, the pointing finger and the middle finger). It states that whenthose first 3 fingers are pointed in a way that they are at right angles to one another, the direction of the magnetic force on a moving charge points in the direction of the middle finger, the thumb points in the direction of the velocity, the pointing finger is in the direction of B.

Practicalizing this, with the directions already stated in the question (velocity is in the positive x direction, the force is in the direction of the acceleration in the positive z-direction), the middle finger is left to point in the negative y-direction.

Using vector notation,

F = qV × B

F = Fx î

qV = qV(z) k

B = Bxî + Byj + Bzk

Doing the cross product,

Fx î = (-By)(qVz)

Since Fx, q, Vz are all positive quantities, By has to be a negative quantity to turn the minus into a plus.

Hence, the magnetic field is directed in the negative y-direction.

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