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.

To find the acceleration of the protons, we can make use of the equation for the centripetal force acting on them.

The centripetal force acting on a charged particle moving in a magnetic field is given by the equation:

F = (mv^2) / r

Where:
F is the centripetal force,
m is the mass of the particle,
v is the velocity of the particle, and
r is the radius of the circular path.

In this case, the protons are moving in a circular path with a radius of 1 meter. As the magnetic field is increasing, the velocity of the protons will also change, resulting in a changing centripetal force.

The relationship between the magnetic field and the velocity of the protons is given by the equation:

F = qvB

Where:
F is the magnetic force,
q is the charge of the protons (1.6 x 10^-19 C),
v is the velocity of the protons, and
B is the magnetic field.

Equating these two expressions for the force, we have:

(mv^2) / r = qvB

Rearranging the equation, we get:

v = (qBr) / m

The rate of change of the magnetic field is given as 0.001 Tesla/second. Let's substitute the given values into the equation to find the acceleration:

v = (qBr) / m
v = (1.6 x 10^-19 C) * (0.001 T/1 s) * (1 m) / (1.67 x 10^-27 kg)
v = 9.58 x 10^6 m/s

So, the velocity of the protons is 9.58 x 10^6 m/s and is increasing. Hence, the acceleration of the protons is positive.

To find the acceleration of the protons in meters/second^2, we need to use the equation for the centripetal acceleration of a charged particle moving in a magnetic field.

The centripetal acceleration of a charged particle moving in a magnetic field is given by the formula:

a = (q * v * B) / m

Where:
a = acceleration of the particle
q = charge of the particle
v = velocity of the particle
B = magnetic field strength
m = mass of the particle

In this case, we are dealing with protons, which have a charge of +e (1.602 × 10^-19 C) and a mass of 1.673 × 10^-27 kg.

The velocity of the protons can be determined using the relationship between the radius of the circular path and the velocity of the particle. The formula for this relationship is:

v = (q * B * r) / m

Where:
v = velocity of the particle
q = charge of the particle
B = magnetic field strength
r = radius of the circular path
m = mass of the particle

Substituting the given values (B = 0.001 T, r = 1 m, q = 1.602 × 10^-19 C, m = 1.673 × 10^-27 kg) into the equation, we can solve for the velocity:

v = (1.602 × 10^-19 C * 0.001 T * 1 m) / (1.673 × 10^-27 kg)

v ≈ 9.58 × 10^6 m/s

Now we can substitute the values for the charge, velocity, magnetic field strength, and mass into the acceleration formula:

a = (1.602 × 10^-19 C * 9.58 × 10^6 m/s * 0.001 T) / (1.673 × 10^-27 kg)

a ≈ 9.56 × 10^16 m/s^2

The acceleration of the protons is approximately 9.56 × 10^16 m/s^2, which means they are speeding up.