Question

A magnetic field existing in a region is given by \(\mathrm{B}^{-}=\mathrm{B}_{0}[1+(\mathrm{x} / \ell)] \mathrm{k} \wedge . \mathrm{A}\) square loop of side \(\ell\) and carrying current I is placed with edges (sides) parallel to \(\mathrm{X}-\mathrm{Y}\) axis. The magnitude of the net magnetic force experienced by the Loop is (a) \(2 \mathrm{~B}_{0} \overline{\mathrm{I} \ell}\) (b) \((1 / 2) B_{0} I \ell\) (c) \(\mathrm{B}_{\circ} \mathrm{I} \ell\) (d) BI\ell

Short answer

The correct answer is not among the given options. The net magnetic force experienced by the square loop is \(3 I \ell B_0\).

Step-by-step solution

Calculate the magnetic field at each side of the square loop

We are given the magnetic field in the region as \(B^{-}=B_0[1+(\frac{x}{\ell})]\,k\). The square loop has sides of length \(\ell\) and is placed parallel to the XY plane, which means the loop has vertices at (0,0), (0, \(\ell\)), (\(\ell\), \(\ell\)), and (\(\ell\), 0). We'll calculate the magnetic field at each side of the loop: Side 1: At x = 0, the magnetic field is \(B_1 = B_0 k\) Side 2: At x = \(\ell\), the magnetic field is \(B_2 = B_0 (1 + 1) k = 2 B_0 k\) Side 3: At x = \(\ell\), the magnetic field is \(B_3 = 2 B_0 k\), as in Side 2. Side 4: At x = 0, the magnetic field is \(B_4 = B_0 k\), as in Side 1.

Determine the magnetic force on each side of the loop

The magnetic force on each side can be determined using the formula \(F = I * l * B * sin(\angle)\). Since the angle between the magnetic field and the loop is 90 degrees, sin(90) = 1. So the magnetic forces on the four sides of the loop can be found using the following calculations: Force on Side 1: \(F_1 = I \ell B_1 = I \ell B_0\) Force on Side 2: \(F_2 = I \ell B_2 = I \ell (2 B_0)\) Force on Side 3: \(F_3 = 0\) (since the current is parallel to the magnetic field, the force is 0) Force on Side 4: \(F_4 = 0\) (similarly, the current is parallel to the magnetic field, the force is 0)

Sum up the magnetic forces for all sides of the loop

To find the net magnetic force experienced by the loop, we sum up the magnetic forces on all sides: Net magnetic force: \(F = F_1 + F_2 + F_3 + F_4 = I \ell B_0 + I \ell (2 B_0) = 3 I \ell B_0\) Now let's compare this result to the given options: a) \(2 \mathrm{~B}_{0} \overline{\mathrm{I} \ell}\) b) \((1 / 2) B_{0} I \ell\) c) \(\mathrm{B}_{\circ} \mathrm{I} \ell\) d) BI\ell None of the options matches our result (3I\(\ell\)B\(_{0}\)); there's a mistake in the given options. Nevertheless, we've shown how to calculate the net magnetic force experienced by the square loop.

Key concepts

Magnetic Field

A magnetic field is a region around a magnetic material where magnetic forces are exerted on moving charges and magnetic dipoles. This force field is denoted by the symbol **B** and is measured in teslas (T). The field can exert a magnetic force on a current-carrying wire placed within it. In many problems like this one, the magnetic field is not uniform, which means its strength and direction can vary at different points.

For the given exercise, the magnetic field is described by the formula:

• \(B = B_0[1 + \frac{x}{\ell}]\,k\)

Where **B** is the magnetic field strength, **B₀** is a constant, **x** is the position along the loop, and **k** is the unit vector along the z-direction. This means that the field has both a constant part **B₀** and a part that increases linearly with **x**.

Understanding the behavior of the magnetic field is crucial to calculating the forces on different sections of the loop.

Current-Carrying Loop

A current-carrying loop is essentially a conductor, such as a wire, bent into a closed shape through which an electric current flows. In this problem, we have a square loop with a side length of \(\ell\), and it carries a current **I**. The shape and orientation of the loop in a magnetic field determine the type and magnitude of the forces experienced by the loop.

The current in the loop interacts with the magnetic field to produce magnetic forces. The direction of these forces depends on the direction of the current and the orientation of the loop with respect to the magnetic field. In this exercise, the loop is oriented parallel to the xy-plane.

• The sides are aligned along the axes.
• The current and magnetic field direction are perpendicular in some sections.

This interaction is a fundamental principle and is described by the right-hand rule in electromagnetism.

Net Magnetic Force

The net magnetic force on a current-carrying loop in a magnetic field is the sum of the forces exerted on each segment of the loop. The direction and magnitude of these individual forces are influenced by the loop's orientation and its current direction.
In our exercise, the force on a segment of the loop is determined by the Lorentz force:

• \(F = I \cdot l \cdot B \cdot \sin\theta\)

Where **F** is the force, **I** is the current, **l** is the segment length, **B** is the magnetic field, and \(\theta\) is the angle between the current and magnetic field (which is 90° here so \(\sin 90° = 1\)).

To find the net force, calculate the forces on each segment and sum them up:

• For sides where the current is perpendicular to **B**, calculate using the strength of the magnetic field at that side.
• For sides where the current runs parallel to **B**, the force is zero due to \(\sin 0 = 0\).

Summing these forces gives the net magnetic force as derived in the solution: **3IℓB₀**. This result doesn't match any given options and suggests an error in those provided options.

Square Loop Geometry

Understanding the geometric configuration of the square loop is vital for solving such problems. The square loop in question has equal-length sides, each denoted by \(\ell\), and vertices located at coordinates (0,0), (0, \(\ell\)), (\(\ell\), \(\ell\)), and (\(\ell\), 0).

• These vertices align the loop parallel to the xy-plane, making calculations of the magnetic force more straightforward.
• Each side of the loop interacts differently with the magnetic field due to differing values of **x** (position along the loop).

This dependency created different magnetic forces at different sections, as outlined in the solution:

• The fields at the sides aligned along the y-axis vary, affecting the force experienced.
• Recognize that geometric arrangement helps predict interaction and simplifies force calculations.

By evenly distributing side lengths and ensuring symmetry, solving for magnetic effects becomes a structured and systematic procedure.