Magnetic Flux

Cyclotron Frequency Independence

In a cyclotron, we want to determine what the frequency of revolution of a charged particle in a magnetic field is independent of.

Step 1: Cyclotron Frequency Formula

The cyclotron frequency ( ) is the frequency at which a charged particle revolves in a uniform magnetic field. It's given by:

Where:

• is the cyclotron frequency.
• is the magnitude of the charge of the particle.
• is the magnitude of the magnetic field.
• is the mass of the particle.

Step 2: Dependence on Energy

From the formula, we can see that the frequency depends on the charge , the magnetic field , and the mass . It does *not* directly depend on the energy of the particle. However, this is true only for non-relativistic speeds. At relativistic speeds, the mass of the particle increases with its energy, and thus the frequency does become energy-dependent. *For the standard introductory physics treatment of the cyclotron, however, we assume non-relativistic speeds*.

Conclusion

Assuming non-relativistic speeds, the frequency of revolution of the charged particle in a magnetic field in a cyclotron is independent of its energy.

Torque and Flux on a Coil in a Magnetic Field

We have a current-carrying coil in a magnetic field B. Torque is τ, θ is the angle between the normal to the plane of the coil and B, and φ is the flux linked with the coil. We need to analyze the relationship between τ, φ, and θ.

Step 1: Torque on a Coil

The torque on a current-carrying coil in a magnetic field is given by:

Where:

• is the torque.
• is the magnetic field strength.
• is the current.
• is the area of the coil.
• is the number of turns in the coil.
• is the angle between the *normal* to the plane of the coil and the magnetic field.

Step 2: Magnetic Flux

The magnetic flux through the coil is given by:

Where:

• is the magnetic flux.
• is the magnetic field strength.
• is the area of the coil.
• is the number of turns in the coil.
• is the angle between the *normal* to the plane of the coil and the magnetic field.

Step 3: Analyze the Case when θ = 90°

When :

• , so the torque is *maximum*. The expression in the original description says that it is minimum, but that is FALSE
• , so the flux is *zero*.

Step 4: Analyze the Case when θ = 0°

When :

• , so the torque is *zero*.
• , so the flux is *maximum*.

Conclusion

Given the choices and having accurately computed everything, the correct choice is if the answer's correct value read: τ is zero and φ is maximum for θ : 0°

Step 1: The magnetic force on a moving charge is given by:
Step 2: Given:
Step 3: Compute the cross product:

Step 4: Expanding the determinant:
Step 5: Compute force:
Step 6: Since represents the direction, the force is N in the direction.
Step 7: Therefore, the correct answer is (E).

Step 1: Paramagnetic materials are characterized by a small positive susceptibility.
Step 2: These materials do not retain magnetization in the absence of an external magnetic field, unlike ferromagnetic materials.
Step 3: According to Curie's Law, the susceptibility of paramagnetic materials is inversely proportional to their temperature.
Step 4: Therefore, as the temperature increases, the magnetic susceptibility decreases, which aligns with option (A).