Question

A square loop of wire of side length \(\ell\) lies in the \(x y\) -plane, with its center at the origin and its sides parallel to the \(x\) - and \(y\) -axes. It carries a current, \(i\), in the counterclockwise direction, as viewed looking down the \(z\) -axis from the positive direction. The loop is in a magnetic field given by \(\vec{B}=\left(B_{0} / a\right)(z \hat{x}+x \hat{z}),\) where \(B_{0}\) is a constant field strength, \(a\) is a constant with the dimension of length, and \(\hat{x}\) and \(\hat{z}\) are unit vectors in the positive \(x\) -direction and positive \(z\) -direction. Calculate the net force on the loop.

Short answer

Answer: The net force on the square loop due to the magnetic field is zero.

Step-by-step solution

Consider one side of the square loop

We will start by analyzing the force on one side of the square loop, say the side parallel to the x-axis with one end on the y-axis. For this side of the loop, the length vector is \(\vec{L}=\ell \hat{x}\). In this region, the magnetic field is : \(\vec{B}=\left(B_{0} / a\right)(z \hat{x}+x \hat{z})\).

Calculate the force on one side of the loop

To find the force on this side of the loop, we can use the expression for the force on a current-carrying wire in the magnetic field: \(\vec{F} = i\vec{L} \times \vec{B}\). Applying the formula: \(\vec{F} = i(\ell \hat{x}) \times \left(B_{0} / a\right)(z \hat{x}+x \hat{z})\). For the cross product, we have: \(\vec{F} = \frac{iB_{0}}{a}\left[\ell \hat{x} \times (z \hat{x}) + \ell \hat{x} \times (x \hat{z})\right]\). Recall that the cross product of two parallel vectors is zero. Also the magnitude of the force is antiproportional to the distance a, because the magnetic field strength decreases as a increases. The relevant term is the one containing cross of \(\hat{x}\) and \(\hat{z}\): \(\vec{F}_1 = \frac{iB_{0}\ell}{a}\left(\ell \hat{x} \times x \hat{z}\right)\).

Analyze the forces on the other sides of the loop

We can apply the same approach to calculate the forces on the other three sides of the square loop. Using symmetry, we will find that the forces on opposite sides of the loop will cancel each other out.

Sum the forces on the loop

Finally, we can sum the forces on all sides of the loop to find the net force on the loop: \(\vec{F}_{net} = \sum_{i=1}^{4} \vec{F}_i = \vec{0}\). Based on the symmetry of the problem, the net force on the square loop is zero. The sum of the four forces on the sides of the loop cancel each other out, leaving no net force on the loop due to the magnetic field.

Key concepts

Lorentz Force

The Lorentz force is a fundamental concept in physics that describes the force exerted on a charged particle moving in a magnetic and/or electric field. It's mathematically expressed as \( \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \), where \( \vec{F} \) is the Lorentz force, \( q \) is the electric charge of the particle, \( \vec{E} \) is the electric field, \( \vec{v} \) is the velocity of the particle, and \( \vec{B} \) is the magnetic field. For a current-carrying loop, like in our exercise, the force is due to the motion of charges within the wire and is given by \( \vec{F} = i(\vec{L} \times \vec{B}) \), where \( i \) is the current and \( \vec{L} \) is the length vector along the wire. This allows us to understand and compute the effects of magnetic fields on circuits and devices.

In the context of the given exercise, the current-carrying loop creates a scenario where the Lorentz force applies to each segment. The goal is to apply the principles governing the Lorentz force to determine the net effect on the loop. By analyzing each segment of the loop individually and using vector cross products, we can understand how forces interact and potentially cancel each other out due to symmetry.

Cross Product in Magnetic Force

The cross product is a mathematical operation used extensively in physics to describe the interaction between vectors in three-dimensional space. In the context of magnetic forces, it forms an essential part of calculating the direction and magnitude of the force experienced by a current-carrying conductor in a magnetic field.

The force on a current element in a magnetic field is given by \( \vec{F} = i(\vec{L} \times \vec{B}) \), where \( \vec{L} \) and \( \vec{B} \) are vectors describing the length direction of the wire and magnetic field, respectively. The cross product here ensures that the resultant force is perpendicular to both \( \vec{L} \) and \( \vec{B} \), a fundamental aspect of how magnetic forces act. In our textbook problem, when calculating \( \vec{F} \), we find that if \( \vec{L} \) and \( \vec{B} \) are parallel, as with some loop segments, the cross product becomes zero, indicating no force is exerted. Understanding the cross product's role helps clarify why certain parts of the loop experience no force.

Magnetic Field

A magnetic field is a vector field that surrounds magnets and electric currents, and it plays a crucial role in determining the behavior of charged particles and currents in its vicinity. The magnetic field is commonly denoted by the symbol \( \vec{B} \) and is measured in teslas (T) in the International System of Units (SI). Magnetic fields are depicted using field lines, with the direction of the magnetic field being tangent to the field lines at any point.

For our exercise, the loop resides within a specified magnetic field \( \vec{B}=(B_{0}/a)(z \hat{x}+x \hat{z}) \). This magnetic field vector has both \( x \) and \( z \) components, implying its orientation changes with position in space. When analyzing the force on the loop, the spatial variation of the magnetic field becomes significant, influencing the force calculation for different segments of the loop. The spatial dependency means that the loop's alignment and position relative to the field source can affect the net force computation.

Symmetry in Physics

Symmetry is a powerful concept in physics, which often simplifies complex problems by showing that certain aspects of a system behave predictably or cancel out due to its geometric or spatial configuration. In physical laws, symmetry can imply conservation laws or invariant properties under transformations such as rotation, reflection, or translation.

In the scenario presented in the exercise, the square loop is symmetrical about its center, and the sides are parallel to the axis of the magnetic field. This setup implies that forces on opposite sides of the loop will be equal in magnitude but opposite in direction, leading to cancellation of forces when summed over the entire loop. Recognizing symmetrical patterns can significantly simplify the task of calculating net forces or fields in a system. As a result, the concept of symmetry helps us deduce that the forces exerted on the square loop by the magnetic field, in this case, result in a net force of \( \vec{0} \), meaning no net force acts on the loop.