Question

A copper wire with density \(\rho=8960 \mathrm{~kg} / \mathrm{m}^{3}\) is formed into a circular loop of radius \(50.0 \mathrm{~cm} .\) The cross-sectional area of the wire is \(1.00 \cdot 10^{-5} \mathrm{~m}^{2}\), and a potential difference of \(0.0120 \mathrm{~V}\) is applied to the wire. What is the maximum angular acceleration of the loop when it is placed in a magnetic field of magnitude \(0.250 \mathrm{~T}\) ? The loop rotates about an axis in the plane of the loop that corresponds to a diameter.

Short answer

The maximum angular acceleration of the copper loop is \(\frac{0.125}{(1.12 \times 10^8)^2} \pi^{-1}~\text{rad/s}^2\).

Step-by-step solution

Calculate the mass of the loop

First, find the volume of the wire, which is equal to the cross-sectional area multiplied by the length of the wire. The length of the wire is equal to the circumference of the circular loop, which is \(2 \pi r\), with \(r = 0.50 m\). Volume of the wire: \(V = A \times l = (1.00 \times 10^{-5}~\text{m}^2)(2 \pi \times 0.50~\text{m}) = 1.00 \times 10^{-5} \times \pi ~\text{m}^3\) Now, calculate the mass of the wire using the density: Mass \(m = \rho V = 8960~\text{kg/m}^3 (1.00 \times 10^{-5} \times \pi ~\text{m}^3) = 8.96 \times 10^7 \pi~\text{kg}\)

Find the moment of inertia of the loop

Since the loop is rotating about an axis in the plane of the loop corresponding to a diameter, we need to find the moment of inertia using the perpendicular axis theorem for a loop of mass m and radius r: \(I = I_x + I_y = \frac{1}{2}mr^2 + \frac{1}{2}mr^2\) Moment of inertia, \(I = mr^2 = (8.96 \times 10^7 \pi~\text{kg})(0.50~\text{m})^2 = 1.12 \times 10^8 \pi~\text{kg}\cdot \text{m}^2\)

Calculate the torque acting on the loop

The torque on the loop due to the magnetic field is determined by the formula: \(\tau = NIBA\sin{\theta}\), where N is the number of turns, I is the current, B is the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the plane of the loop. Here, N = 1, A = \(\pi r^2\), θ = 90° (\(\sin{90} = 1\)). The current, I, can be determined using Ohm's law, \(V=IR\), and the resistance of the wire, \(R\), can be calculated using the formula, \(R = \frac{\rho l}{A}\). Resistance: \(R = \frac{8.96 \times 10^7~\text{Ω}\cdot \text{m}^-1 \times 2 \pi \times 0.50 ~\text{m}}{1.00 \times 10^{-5} ~\text{m}^2} = 1.12 \times 10^8 \pi~\text{Ω}\) Current: \(I = \frac{V}{R} = \frac{0.0120~\text{V}}{1.12 \times 10^8 \pi~\text{Ω}} = \frac{1}{\pi\times10^8}\) A Now, calculate the torque: \(\tau = NIBA = \left(\frac{1}{\pi\times10^8}\text{A}\right)(0.250~\text{T})(\pi\times 0.50^2~\text{m}^2)=\frac{0.125}{10^7}\) Nm

Calculate the angular acceleration of the loop

Using the formula \(\alpha = \frac{\tau}{I}\), we can find the angular acceleration: Angular acceleration: \(\alpha = \frac{\tau}{I} = \frac{\frac{0.125}{10^7}~\text{Nm}}{1.12 \times 10^8 \pi ~\text{kg}\cdot \text{m}^2} = \frac{0.125}{(1.12 \times 10^8)^2} \pi^{-1}~\text{rad/s}^2\) Now you have the maximum angular acceleration of the copper loop when placed in a magnetic field of magnitude 0.250 T.

Key concepts

Magnetic Torque

Understanding how forces act in magnetic fields is crucial, especially when dealing with rotating objects. Magnetic torque is a force that can cause an object to rotate when placed in a magnetic field. It is calculated using the formula
\( \tau = NIBA\text{sin}\theta \),
where \( \tau \) is the torque, \( N \) is the number of turns in the coil, \( I \) is the current in amps, \( B \) is the magnetic field in teslas, \( A \) is the area of the loop, and \( \theta \) is the angle between the field and the normal to the loop.
The torque is maximized when the field is perpendicular to the plane of the loop, as \( \text{sin}\theta = 1 \) when \( \theta = 90° \). In our example with the copper wire, the magnetic torque is what initiates the loop's angular acceleration in the magnetic field.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation rate. It depends not only on the mass but also on how that mass is distributed with respect to the axis of rotation.
For a rotating object, like our circular loop of wire, the moment of inertia is crucial for determining the angular acceleration. Using the perpendicular axis theorem, which states that the moment of inertia around an axis in the plane (\( I_z \)) is equal to the sum of the moments of inertia through two perpendicular axes that intersect the first axis (\( I_x \) and \( I_y \)), we can find the loop's moment of inertia with respect to its diameter.
The formula for a thin circular loop is simplified to \( I = mr^2 \), when rotating about a diameter, where \( m \) is the mass of the loop and \( r \) is its radius. With the moment of inertia, we then can proceed to calculate how fast the loop can accelerate in a magnetic field.

Ohm’s Law

Ohm's Law is a foundational principle in the study of electricity, describing the relationship between voltage, current, and resistance in an electrical circuit. It's defined by the equation
\( V = IR \),
where \( V \) is the voltage across a resistor or a loop section, \( I \) is the current flowing through it, and \( R \) is the resistance. This law holds true for many materials, including the copper wire in our exercise.
By knowing the applied voltage and the resistance of the wire, we can determine the current that flows through our circular loop. The resistance, in turn, can be determined using the material's resistivity, length of the wire, and its cross-sectional area. The current is then used to calculate the magnetic torque that influences the loop's angular acceleration.

Perpendicular Axis Theorem

The perpendicular axis theorem is a mathematical tool that aids in calculating an object's moment of inertia. It states that the moment of inertia of a planar object about an axis perpendicular to its plane (that is passing through its center of mass) is equal to the sum of the moments of inertia of the object about two mutually perpendicular axes lying in the plane of the object. For instance,
\( I_z = I_x + I_y \),
where the z-axis is perpendicular to the plane of the object, and the x and y-axes lie in the plane.
In practice, this theorem simplifies computations, especially in cases like our circular loop, allowing us to determine the rotational properties of an object without complex integration. By applying this theorem, we easily obtained the loop's moment of inertia about the axis of rotation—key information needed for calculating its angular acceleration in the magnetic field.