Question

A point charge with a mass of 1.81 ng and a charge of +1.22 uC moves in the x-y plane with a velocity of 3.00 x 104 m/s in a direction 15° above the +x-axis. At time t=0, the point charge enters a uniform magnetic field of strength 1.25 T that points in the +x-direction. Assume that the point charge remains immersed in the uniform magnetic field after time t=0.

a. (5 points) What is the magnitude and direction of the magnetic force that the magnetic field exerts on the point charge at time t=0?

b. (5 points) How does the x-component of the charge’s initial velocity effect the motion of this point charge as it moves through the uniform magnetic field? Does its magnitude change? Does its direction change? Explain your reasoning.

c. (5 points) How does the y-component of the charge’s initial velocity effect the motion of this point charge as it moves through the uniform magnetic field? Does its magnitude change? Does its direction change? Explain your reasoning.

d. (5 points) Use your answers from parts 1b & 1c to explain why the path of this point charge is helical (corkscrew-shaped). Explain your reasoning.

e. (2.5 points) Determine the radius of the circular part of the point charge’s helical path.

Answer 1

a)

q = charge = 1.22 x 10^-6 C

representing the velocity in unit vector notation where "i" denotes direction along x-axis and "j" denotes direction along Y-axis.

= velocity of the charge = ((3 x 10⁴) Cos15) i + ((3 x 10⁴) Sin15) j = (2.89 x 10⁴) i + (0.78 x 10⁴) j

= magnetic field = 1.25 i T

magnetic force on the charge is given as

= q (x )

= (1.22 x 10^-6) (((2.89 x 10⁴) i + (0.78 x 10⁴) j) x (1.25 i))

=  (1.22 x 10^-6) (9.75 x 10³) (- k)

= 0.012 (- k) N

magnitude of the force = 0.012 N

direction = negative z-direction

b)

the x-component of velocity of charge being in same direction as the magnetic field, remain unaeffected. its magnitude and direction remains same since magnetic force does not apply force in its direction.

c)

the y-component of velocity experience acceleration due to magnetic force . this acceleration being perpendicular to the direction of component of velocity in Y-direction, only change the direction and does not affect the magnitude of component of velocity in y-direction . its direction change and the charge moves in a circle.

d)

due to magnetic force on y-component of the velocity , the charge moves in circle. and the x-component of the velocity moves it along the X-direction. hence the charge moves in circle as well as moves forward simultaneously. this make the path helical

e)

radius of the circular path is given as

r = m v Sin/(qB)

r = (1.81 x 10^-9 x 10^-3) (3 x 10⁴) Sin15 /((1.22 x 10^-6) (1.25))

r = 9.2 x 10^-3 m