Consider free protons following a circular path in a uniform magnetic field with a radius of 1 meter. At t = 0, the magnitude of the uniform magnetic field begins to increase at 0.001 Tesla/second. Enter the tangential acceleration of the protons in meters/second^2: positive if they speed up and negative if they slow down.
The magnetic field itself will produce no tangential acceleration. But then, consider Maxwell's equation, or more precisely, the resulting electric field created by the changing magnetic field, which will move the proton in the tangential direction.
By using Faraday's law E= - r/2 x dB/dt so a=E x q/m= -r/2 x dB/dt x q/m so a=- 5.22*10^4m/s^2
ou should use Faraday's law of induction to find the electric field E at distance r from the center and from that the tangential acceleration due to F=Eq=m*atan
but E= - r/2 x dB/dt
so a=E x q/m=
a= -r/2 *dB/dt*q/m so
a=-1/2*.001*1.60E−19/1.67E-27
a=-479m/s^2
Honestly, this does not seem right, so check it.
To find the tangential acceleration of the protons, we need to use the equation for the magnetic force acting on a particle moving in a magnetic field.
The magnetic force on a charged particle moving in a magnetic field is given by the equation:
F = qvBsinθ
Where:
- F is the magnetic force
- q is the charge of the particle
- v is the velocity of the particle
- B is the magnetic field strength
- θ is the angle between the velocity vector and the magnetic field vector
In this case, the protons are moving in a circular path, so their velocity can be represented as v = ωr, where ω is the angular velocity and r is the radius of the circular path.
The tangential acceleration can be found using the relation:
a_t = (dv/dt)r = (dω/dt)r
Since the magnitude of the uniform magnetic field B is increasing with time, the angle θ between the velocity vector and the magnetic field vector is changing. Thus, there will be a change in the tangential acceleration of the protons.
To find the tangential acceleration, we need to differentiate the equation v = ωr with respect to time:
dv/dt = d(ωr)/dt
dv/dt = r(dω/dt)
Now, let's substitute this into the equation for tangential acceleration:
a_t = (dω/dt)r
We were given that the magnitude of the magnetic field is increasing at a rate of 0.001 Tesla/second at t = 0. So we can substitute this value into the equation for the tangential acceleration:
a_t = (0.001)(1) = 0.001 meters/second^2
Therefore, the tangential acceleration of the protons is 0.001 meters/second^2.
Note: The tangential acceleration is positive because the magnitude of the magnetic field is increasing, causing the protons to speed up.