Question

A small charged aluminum ball with a mass of \(3.435 \mathrm{~g}\) is moving northward at \(3183 \mathrm{~m} / \mathrm{s}\). You want the ball to travel in a horizontal circle with a radius of \(1.893 \mathrm{~m}\) and in a clockwise direction when viewed from above. Ignoring gravity, the magnitude of the magnetic field that must be applied to the aluminum ball to cause it to move in this way is \(B=0.5107 \mathrm{~T}\). What is the charge on the ball?

Short answer

Answer: The approximate charge on the small charged aluminum ball is \(1.2389 \times 10^{-6}\, C\).

Step-by-step solution

Write the centripetal force equation and the magnetic force equation on the charged particle.

The centripetal force acting on a particle with mass \(m\) and velocity \(v\) moving in a circle of radius \(r\) is given by: \(F_{c} = m\frac{v^2}{r}\) The magnetic force acting on a particle with charge \(q\) moving with velocity \(v\) in a magnetic field \(B\) is given by: \(F = q(vB\sin(\theta))\)

Equate the centripetal force and the magnetic force acting on the ball

Since the centripetal force holding the particle in a circular orbit is the magnetic force acting on it, we can write: \(m\frac{v^2}{r} = q(vB\sin(\theta))\)

Express the charge q in terms of the other variables and known values.

We can rearrange the above equation to solve for the charge \(q\): \(q = \frac{m\frac{v^2}{r}}{vB\sin(\theta)}\) We are given the mass \(m = 3.435\, g = 3.435\cdot 10^{-3}\, kg\), the velocity \(v = 3183\, m/s\), the radius \(r = 1.893\, m\), and the magnitude of the magnetic field \(B = 0.5107\, T\). Since the magnetic field is applied vertically, the angle between the particle's velocity vector and the magnetic field is \(\theta=90^\circ\) or \(\sin(\theta) = 1\). Thus, we can plug in these values to find the charge:

Plug in the values and calculate the charge q.

Substitute the known values into the equation for \(q\): \(q = \frac{(3.435\cdot 10^{-3}\, kg)\frac{(3183\, m/s)^2}{1.893\, m}}{(3183\, m/s)(0.5107\, T)(1)}\) Calculate the charge \(q\): \(q \approx 1.2389 \times 10^{-6}\, C\) The charge on the aluminum ball is approximately \(1.2389 \times 10^{-6}\, C\).

Key concepts

Centripetal Force

Imagine you are swinging a ball on a string in a circular path. The force that keeps the ball moving in that circle, rather than flying off in a straight line, is what we call the centripetal force. This invisible tether is not an actual force but rather the result of other forces that direct the object to follow a curved path.

In physics, the mathematical expression for centripetal force when an object of mass m is moving at a velocity v in a path with radius r is represented as \(F_{c} = m\frac{v^2}{r}\). This equation tells us the amount of force required to keep the object moving in a circle. The faster the object moves or the tighter the circle (smaller radius), the more centripetal force that’s needed. When the centripetal force is no longer applied, the object will move in a straight line due to inertia, following Newton's first law of motion.

Magnetic Field

The area around a magnet where magnetic forces can be detected is called the magnetic field. It is visualized by drawing lines that extend from the north pole of a magnet to its south pole. These invisible lines represent the direction and strength of the forces that would act on iron filings or other magnets placed within the field.

The magnetic field strength is measured in teslas (T) and it can exert a magnetic force on moving charged particles, such as electrons or ions. When a charged particle moves through a magnetic field, it will experience a force that is perpendicular to both its velocity and the field direction. This fundamental interaction is essential for technologies like MRI machines and is the basis for the operation of electric motors and generators, which convert between mechanical energy and electrical energy.

Charge of a Particle

The charge of a particle is a fundamental property that determines how it will interact with electric and magnetic fields. Charged particles are classified as either positive or negative, which influences the direction of the force they experience in a magnetic field. For instance, protons have a positive charge, while electrons have a negative charge.

The unit of electric charge is the coulomb (C), which represents a considerable amount of charge. In practice, charges on particles are often tiny fractions of a coulomb but can accumulate significant effects. The charge is crucial because it’s directly proportional to the magnetic force exerted on the particle when it moves across a magnetic field. In other words, the stronger the charge, the greater the force exerted.

Motion in a Magnetic Field

The motion of a charged particle in a magnetic field is fascinating because it is perpendicular to both the direction of the particle's velocity and the field lines. This motion is often circular or helical, depending on whether there’s an additional component of velocity parallel to the field.

The force experienced by a charged particle in a magnetic field, referred to as the Lorentz force, is given by the equation \(F = q(vB\sin(\theta))\), where q is the charge, v is the velocity, B is the magnetic field strength, and \(\theta\) is the angle between the velocity and the magnetic field direction. A perfect example of this is when particles are shunted into a circular path by a uniform magnetic field, an occurrence that can be observed in particle accelerators and auroras around the magnetic poles of Earth.