A charged particle enters a uniform magnetic field with a velocity vector at an angle of 45° with the magnetic field. The pitch of the helical path followed by the particle is p. The radius of the helix will be

The path of a charged particle in a magnetic field is a helix if the velocity of the particle makes an angle with the magnetic field. The pitch (p) of the helix is the distance the particle moves along the direction of the magnetic field in one complete cycle. The radius (r) of the helix is the perpendicular distance from the axis to the particle.

The radius of the helix is given by the formula:

r = mv⊥/qB

where:

• m is the mass of the particle,
• v⊥ is the component of the velocity perpendicular to the magnetic field,
• q is the charge of the particle, and
• B is the magnetic field.

Given that the velocity vector makes an angle of 45° with the magnetic field, the component of the velocity perpendicular to the magnetic field is v⊥ = v sin(45°) = v/√2.

Substituting this into the formula for the radius gives:

r = mv/(qB√2)

The pitch of the helix is given by the formula:

p = 2πmv∥/qB

where v∥ is the component of the velocity parallel to the magnetic field. Given the 45° angle, v∥ = v cos(45°) = v/√2.

Substituting this into the formula for the pitch gives:

p = 2πmv/(qB√2)

Comparing the expressions for r and p, we see that they are equal. Therefore, the radius of the helix is equal to the pitch.