Question

Prove that the time for one revolution of a charged particle moving perpendicular to a uniform magnetic field is independent of its speed. (This is the principle on which the cyclotron operates.) In doing so, write an expression that gives the period \(T\) (the time for one revolution) in terms of the mass of the particle, the charge of the particle, and the magnetic field strength.

Short answer

If not, what factors determine the time for one revolution? Answer: No, the time for one revolution of a charged particle moving perpendicular to a uniform magnetic field does not depend on the particle's speed. The period of one revolution (T) is determined only by the mass of the particle (m), the charge of the particle (q), and the magnetic field strength (B). The expression for the period is given by: $$ T = \frac{2 \pi m}{q \cdot B} $$

Step-by-step solution

Identify Lorentz force acting on charged particle

The Lorentz force acting on a charged particle moving in a magnetic field is given by the following formula: $$ F = q \cdot v \cdot B $$ Where \(F\) is the magnitude of the Lorentz force, \(q\) is the charge of the particle, \(v\) is the particle's speed, and \(B\) is the magnetic field strength.

Find the centripetal force acting on the charged particle

In circular motion, the centripetal force is given by the formula: $$ F_c = \frac{m \cdot v^2}{r} $$ Where \(F_c\) is the magnitude of the centripetal force, \(m\) is the mass of the particle, and \(r\) is the radius of the circular path. Since the Lorentz force acts as the centripetal force in this case, we have: $$ q \cdot v \cdot B = \frac{m \cdot v^2}{r} $$

Express the radius of the circular path in terms of speed and other parameters

Solve for r in the equation found in Step 2: $$ r = \frac{m \cdot v}{q \cdot B} $$

Relate the circumference of the circular path to the period of motion

The distance traveled by the charged particle in one full revolution is the circumference of the circular path, which is given by the formula: $$ C = 2 \pi r $$ The time taken for one complete revolution, or the period of the motion (\(T\)), is the distance traveled divided by the speed: $$ T = \frac{C}{v} = \frac{2 \pi r}{v} $$

Find the expression for the period of motion in terms of particle mass, charge, and magnetic field strength

Insert the expression for \(r\) found in Step 3 into the equation for \(T\) found in Step 4: $$ T = \frac{2 \pi \cdot \frac{m \cdot v}{q \cdot B}}{v} $$ Simplify the expression by canceling the \(v\) term, as our goal is to show that the period is independent of the particle's speed. The final expression for the period is: $$ T = \frac{2 \pi m}{q \cdot B} $$ This expression shows that the period (\(T\)) for one revolution of a charged particle moving perpendicular to a uniform magnetic field is independent of the particle's speed, as it does not contain any term related to the speed \(v\). The period is determined solely by the mass of the particle (\(m\)), charge of the particle (\(q\)), and the magnetic field strength (\(B\)).