Question

A particle of given mass m 1kg, and charge q1 Couloumb, is placed in a magnetic field B 1k Tesla and thus is subjected to the effects of the Lorentz force. The force causes it to have an acceleration: a =-(v × B), where v is the particle's velocity. Assume that the motion of the particle remains in the x-y plane and neglect the gravity. If at t-0 sec, the particle was at x y 0 m with an initial speed voー10 ^ m/s, find the equation of the path traced by the particle as it moves. Plot this path. Hint: What is the direction and magnitude of the v crossed with B? If along with the Lorentz force, there was also a drag force acting on the particle that causes an additional acceleration a =-cu, where the coefficient c 0.1 Ns/m, what do you think the path will look like, approximately? Give your reasoning. Once again assume at t -0 sec, the particle was at x-y 0 m with an initial speed vo - 10î m/s. We don't want you to derive the equation of the path in this case, we just want you to reason what its shape will look like

Answer 1

a) y = -0.05*x^2

b) Range shorter, Radius of circular/spiral motion shorter

Step-by-step explanation:

Given:

- The charge q = 1 C

- Magnetic Field = 1 k Tesla

- Mass of the charge m = 1 kg

- Acceleration due to Lorentz Force a = q ( v x B ) / m

- Acceleration due to Drag Force a = - c*v

- The velocity of the charge @ x = y = 0, v_i = 10 i

Find:

- Equation of path of the charge in the given magnetic field

- Describe the path with addition of Drag Force.

Solution:

- First we will determine the magnitude and direction of the acceleration due to Lorentz Force as follows:

a = q ( v x B ) / m

- Input values:

a = 1 ( 10 i x 1 k ) / 1 = -10 j

- We see that the acceleration acts in the negative y direction.

- To determine the path of the charge, we will use kinematic equation of motion for the particle as follows:

x = v_i*t

y = 0.5*a*t^2

- Determine the parametric equations for displacements in x and y directions:

x = 10*t

y = -0.5*10*t^2

Combine the parametric equations to determine the equation of path followed by the charge:

t = x / 10

y = -0.5*x^2 / 100

y = -0.05*x^2

- For the second case we have the acceleration due to drag force relation:

a = - 0.1*v_i

a = -0.1*10 i

a = - i

- There is a deceleration component of drag force acting in x direction, The parametric equation would be as follows:

x = 10*t - 0.5*1*t^2

x = 10*t - 0.5*t^2

- From this we can see that the x coordinate increases at a decreasing rate. Hence, the range of the projectile like motion will decrease. The total amount of distance traveled by the charge in x direction will decrease.

-Hence, after sufficient amount of time the charge moves in a circular motion. The radius of the circular motion will be shorter as compared to when we neglect the acceleration due to drag force.