Question

An alpha particle \(\left(m=6.64 \cdot 10^{-27} \mathrm{~kg}, q=+2 e\right)\) is accelerated by a potential difference of \(2700 . \mathrm{V}\) and moves in a plane perpendicular to a constant magnetic field of magnitude \(0.340 \mathrm{~T}\), which curves the trajectory of the alpha particle. Determine the radius of curvature and the period of revolution.

Short answer

Answer: The radius of curvature is 0.014 meters and the period of revolution is 1.22 x 10^-7 seconds.

Step-by-step solution

Find the velocity of the alpha particle

We will use the formula: \(eV = \frac{1}{2}mv^2\) Solve for \(v\): \(v = \sqrt{\frac{2eV}{m}}\) Using the given values, \(m = 6.64 \times 10^{-27} kg\), \(q = +2e\), and \(V = 2700 V\), we have: \(v = \sqrt{\frac{2(2 \times 1.6 \times 10^{-19} C)(2700 V)}{6.64 \times 10^{-27} kg}} = \sqrt{\frac{8.64 \times 10^{-16} J}{6.64 \times 10^{-27} kg}}\) \(v = 1.16 \times 10^7\: m/s\)

Calculate the radius of curvature

Now, we use the formula for the radius of curvature \(r = \frac{mv}{qB}\): \(r = \frac{(6.64 \times 10^{-27} kg)(1.16 \times 10^7\: m/s)}{(2 \times 1.6 \times 10^{-19} C)(0.340\: T)}\) \(r = 0.014\: m\) The radius of curvature is 0.014 meters.

Find the period of revolution

Using the period of revolution formula \(T = \frac{2\pi m}{qB}\), we can find the period of revolution: \(T = \frac{2\pi (6.64 \times 10^{-27} kg)}{(2 \times 1.6 \times 10^{-19} C)(0.340\: T)}\) \(T = 1.22 \times 10^{-7}\: s\) The period of revolution is \(1.22 \times 10^{-7}\) seconds. Therefore, the radius of curvature of the alpha particle's trajectory is 0.014 meters and the period of revolution is \(1.22 \times 10^{-7}\) seconds.

Key concepts

Radius of Curvature

The radius of curvature is fundamental to understanding the motion of charged particles in a magnetic field. It's a measure of the 'curvedness' of the path that a moving charged particle follows in the presence of a magnetic field. Essentially, it tells us how tightly an alpha particle, like the one in our exercise, is bending while it whizzes through space.

Let's consider an alpha particle zooming straight into a magnetic field that suddenly gives it a nudge sideways. It doesn't go in a straight line anymore but bends into a circle. The radius of this circle, which comes from the center to any point on the circumference where our particle lies, is the radius of curvature. The smaller the radius, the tighter the curve.

In mathematical terms, it's illustrated using the formula: \( r = \frac{mv}{qB} \) where \(m\) represents the mass of the particle, \(v\) its velocity, \(q\) its charge, and \(B\) the magnetic field strength. From the exercise, you can see that once we determined the alpha particle's velocity, calculating the radius of curvature was simply a matter of substituting the values.

Period of Revolution

The period of revolution is another critical concept when describing circular motion of charged particles in a magnetic field. It's the time taken for a particle to complete one full orbit along its circular path. For an alpha particle in a uniform magnetic field, as long as the speed and the magnetic force are constant, this period remains the same for each revolution.

The formula to calculate the period of revolution is: \( T = \frac{2\pi m}{qB} \) This equation tells us that the period of revolution relies on the mass of the particle, its charge, and the magnetic field's intensity. The fascinating thing about the period of revolution is that it’s independent of the particle’s velocity, which means even if our alpha particle speeds up or slows down, the period stays constant as long as the charge to mass ratio is the same and the magnetic field doesn't change. In our exercise, the values plugged into this formula led us to the alpha particle's period of revolution, which is a crucial piece of information for understanding its motion dynamics.

Charge-to-Mass Ratio

The charge-to-mass ratio (\(q/m\)) is a way to characterize how much charge a particle carries compared to its mass. This ratio is instrumental in the study of particle physics because it determines how a particle will respond to electric or magnetic fields.

An example is in the motion of our alpha particle. The behavior of the particle in the magnetic field – how it curves, how fast it spins around in a circle – depends on this charge-to-mass ratio. In both formulas used for the radius of curvature and the period of revolution, the ratio pops up as part of the denominator. This shows that a higher charge or a lower mass would result in a greater force on the particle and, thus, a tighter curve and quicker orbit.

In practice, scientists have used measurements of the charge-to-mass ratio to identify and characterize subatomic particles, making it a cornerstone in the field of particle physics. The charge-to-mass ratio helps us predict and understand the trajectories of charged particles under the influence of electromagnetic fields.

Uniform Magnetic Field

A uniform magnetic field is a field where the magnetic force is the same in both magnitude and direction at any point within the field. This uniformity is key for simplifying calculations about the motion of charged particles, such as our alpha particle.

In our context, the uniform magnetic field applies a constant force perpendicular to the velocity of the particle, causing it to move in a circular path. The force experienced by the particle depends on its velocity, charge, and the magnetic field's strength according to the formula: \( F = qvB \) where again, \(q\) is the charge of the particle, \(v\) is its velocity, and \(B\) is the strength of the magnetic field.

It's important to note that a non-uniform magnetic field would result in a more complex path, and possibly a changing radius of curvature and pace of revolution. But the beauty of a uniform magnetic field lies in its predictability, making it much easier to deduce how particles will behave within it.