Question

A square loop of wire with a side length of \(10.0 \mathrm{~cm}\) carries a current of 0.300 A. What is the magnetic field in the center of the square loop?

Short answer

Answer: The magnetic field at the center of the square loop is approximately \(7.08 \times 10^{-7}\ \mathrm{T}\).

Step-by-step solution

Understanding the Biot-Savart law

The Biot-Savart law states that the magnetic field dB created by an infinitesimal segment of a current-carrying wire with length element dl, carrying current I, is given by: $$dB = \frac{\mu_0 I}{4 \pi} \frac{dl \times \hat{r}}{r^2}$$ where \(\mu_0\) is the permeability of free space, \(\hat{r}\) is the unit vector pointing from the length element to the point where the magnetic field is being considered, and r is the distance from the length element to the point.

Applying the Biot-Savart law to a single side of the square

Since the problem has symmetry, the magnetic field contribution of each side at the center can be assumed to be equal. We will consider one side and then multiply its contribution by 4. Let's consider one side of the square with length L (\(10.0\ \mathrm{cm} = 0.1\ \mathrm{m}\)) and current I (\(0.300\ \mathrm{A}\)). We will use the Biot-Savart law to find the magnetic field dB created by a length element dl along the side at the center of the square (distance L/2 from one end of the side). For the chosen side, the angle between dl and r will be 45°, and their cross product magnitude will be given by |dl||r|\(\sin{\theta}\) = dl(\(\frac{L}{2}\)). Therefore, the magnetic field contribution from this element at the center will be: $$db = \frac{\mu_0 I}{4 \pi} \frac{dl(\frac{L}{2})}{(\frac{L}{2})^2+L^2}$$

Integrate the magnetic field over the entire side

To find the total magnetic field contribution of one side at the center, we need to integrate the above expression from one end of the side to the other: $$B_{side} = \frac{\mu_0 I (\frac{L}{2})}{4 \pi} \int_{0}^{L} \frac{1}{(\frac{L}{2})^2+L^2}$$ The integral becomes a constant, so it can be simplified: $$B_{side} = \frac{\mu_0 I (\frac{L}{2})}{4 \pi ((\frac{L}{2})^2+L^2)} \times L$$

Sum up contributions from all sides

Since all sides are equal, we can find the net magnetic field at the center by multiplying the contribution of one side by 4: $$B_{center} = 4 B_{side}$$ $$B_{center} = 4 \times \frac{\mu_0 I (\frac{L}{2})}{4 \pi ((\frac{L}{2})^2+L^2)} \times L$$ Now, we will substitute the given values and constants (\(\mu_0 = 4\pi \times 10^{-7} \mathrm{T\cdot m/A}\), I = 0.300 A, and L = \(0.1\ \mathrm{m}\)): $$B_{center} = 4 \times \frac{(4\pi \times 10^{-7} \mathrm{T\cdot m/A})(0.300\ \mathrm{A})(0.05\ \mathrm{m})}{4 \pi ((0.05\ \mathrm{m})^2+(0.1\ \mathrm{m})^2)} \times 0.1\ \mathrm{m}$$

Calculate the magnetic field at the center

Now solve for the magnetic field at the center: $$B_{center} = 7.08 \times 10^{-7}\ \mathrm{T}$$ Therefore, the magnetic field in the center of the square loop is approximately \(7.08 \times 10^{-7}\ \mathrm{T}\).

Key concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle that explains how currents generate magnetic fields. In essence, it states that an electric current in a wire produces a magnetic field that encircles the wire. The direction of this magnetic field follows the right-hand rule, which means that if you point the thumb of your right hand in the direction of the current, your curled fingers indicate the direction of the magnetic field.

Mathematically, the Biot-Savart Law is expressed as \, where \(dB\) represents the infinitesimal magnetic field produced by a segment of wire with current \(I\), \(dl\) is an infinitesimal length of the wire, \(\hat{r}\) is a unit vector pointing from the wire to the point where the magnetic field is measured, \(r\) is the distance between the wire and the point, and \(\mu_0\) is the permeability of free space. This law is essential for calculating the magnetic field at any point in space due to a current-carrying conductor.

Permeability of Free Space

Permeability of free space, represented by \(\mu_0\), is a physical constant that describes how a magnetic field can penetrate the vacuum of free space. In the SI units, it is defined as \(4\pi \times 10^{-7} \mathrm{T\cdot m/A}\) (Tesla meters per Ampere). The value of the permeability of free space is crucial because it sets the strength of the magnetic field produced by a current as explained in the Biot-Savart law. Furthermore, it relates the electric and magnetic fields to the speed of light, as \(c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}\), where \(c\) is the speed of light and \(\epsilon_0\) is the permittivity of free space.

Magnetic Field Contribution

In problems involving symmetry and magnetic fields, the contribution from symmetric elements can be combined to find the total magnetic field. For example, when calculating the magnetic field at the center of a square loop, each side of the loop contributes equally to the total field at the center due to symmetry.

By assessing one segment and understanding its behavior, we can multiply its effect by the number of identical segments to get the total magnetic field. It's important to note that this relies on symmetry; if the loop weren't square or the current distribution wasn't uniform, the situation would be more complex and would require a different approach.

Integration of Magnetic Field

The integration of the magnetic field across the length of a wire segment is a mathematical approach to sum up all the infinitesimal contributions to the field. When applying the Biot-Savart law to a finite length of wire, integration over the length of the wire is necessary because the magnetic field varies at different points along the length of the wire. This process involves setting up the integral with the appropriate limits, in our example from 0 to \(L\), the length of one side of the square.

Integration allows us to get the total magnetic field from the entire wire segment, considering all the contributions added up over the space. In the case of the square loop, the result of the integration gives the magnetic field due to one entire side of the square, and this value is then used to calculate the net field at the center of the loop by considering all four sides.