In earlier learning sequences we described how a static magnetic field cannot change the speed (and therefore kinetic energy) of a free charged particle. A changing magnetic field can, and this is one way particle beams are accelerated. 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 the acceleration of the protons in meters/second^2: positive if they speed up and negative if they slow down

Question

In earlier learning sequences we described how a static magnetic field cannot change the speed (and therefore kinetic energy) of a free charged particle. A changing magnetic field can, and this is one way particle beams are accelerated. 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 the acceleration of the protons in meters/second^2: positive if they speed up and negative if they slow down

Answer 1

A beam of electrons moves at right angles to a magnetic field of 4.5 × 10-2 tesla. If the electrons have a velocity of 6.5 × 106 meters/second, what is the force acting on the electrons? The value of q = -1.6 × 10-19 coulombs

Answer 2

To find the acceleration of the protons, we need to use the formula for the centripetal acceleration of a charged particle moving in a magnetic field:

a = (q * v * B) / m

Where:
a = acceleration of the protons
q = charge of the protons (in Coulombs)
v = velocity of the protons (in meters/second)
B = magnetic field strength (in Tesla)
m = mass of the protons (in kilograms)

Since the protons are following a circular path, the velocity can be calculated using the formula for the speed of a particle moving in a circular path:

v = (2 * π * r) / T

Where:
v = velocity of the protons
r = radius of the circular path (in meters)
T = time period for one complete revolution (in seconds)

First, let's calculate the velocity of the protons. The time period can be determined by dividing the circumference of the circular path by the velocity:

Circumference = 2 * π * r
T = Circumference / v

Given that the radius is 1 meter, the circumference is 2 * π * 1 = 2π meters.

Now, we need to calculate T:

T = (2π meters) / v

Next, we can substitute this value of T in the equation for v and solve for v:

v = (2 * π * r) / T

Substituting the values, we find:

v = (2 * π * 1) / [(2π meters) / v]
v = (2 * π * v) / (2π meters)
v = v meters/second

Since v appears on both sides, this equation tells us that the velocity will remain constant.

Now, let's calculate the acceleration using the formula mentioned earlier:

a = (q * v * B) / m

Since the velocity is constant, the acceleration will be zero:

a = (q * v * B) / m
a = (q * v * B) / m
a = (q * v * B) / m
a = 0 meters/second^2

Therefore, the protons will neither speed up nor slow down since the acceleration is zero.