Your approach is generally on the right track, but there are a few points of clarification to address regarding the relation between the side length $l$ and the torque radius $r$, as well as a few minor adjustments in the derivation to reach the desired result.

Let's analyze your steps closely.

Understanding the Torque Formula

• You correctly started by relating the magnetic flux density $\mathbf{B}$ to the magnetic field $\mathbf{H}$ using: $\mathbf{B} = \mu_0 \mathbf{H}$ where $\mu_0$ is the permeability of free space (not permittivity, which refers to $\epsilon_0$).

• The force on a current-carrying wire in a magnetic field is given by: $\mathbf{F} = I_0 \mathbf{l} \times \mathbf{B}$ or equivalently: $F = I_0 l \mu_0 H_0 \sin(\theta)$ where $\theta$ is the angle between the current element and the magnetic field.

Torque Expression

Torque $\mathbf{\tau}$ is given by: $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$ For a rectangular or square loop, the torque is maximized when the loop is oriented perpendicular to the magnetic field (i.e., $\theta = 90^\circ$, giving $\sin(\theta) = 1$).

Key Issue: Relation Between $l$ and $r$

Here’s the important part:

• $l$ is the side length of the square loop (which is given as 5 cm or 0.05 m).
• $r$ represents the "lever arm" or distance from the axis of rotation to the point where the force is applied. For a square loop, this is typically half the side length, i.e.: $r = \frac{l}{2}$ This is because the torque is generated about the center of the loop, and the force acts on the sides at a distance $\frac{l}{2}$ from the axis of rotation.

Correcting the Torque Expression

Given the proper relation for $r$, the torque is expressed as: $\tau = r I_0 l \mu_0 H_0 \sin(\theta)$ Substitute $r = \frac{l}{2}$: $\tau = \frac{l}{2} I_0 l \mu_0 H_0 \sin(\theta)$ At maximum torque, $\sin(\theta) = 1$, so: $\tau_{\text{max}} = \frac{l^2}{2} I_0 \mu_0 H_0$ This matches the form given in the problem statement up to a factor of $\frac{1}{2}$, and since the required expression was $I_0 l^2 \mu_0 H_0$, this suggests that the model considers the torque contributions from multiple segments of the loop.

Summary of Corrections:

1. $r$ (the effective lever arm) should be $\frac{l}{2}$, as the force acts at a distance half the side length of the square loop from the axis of rotation.
2. When calculating torque, you need to account for this factor of $\frac{1}{2}$, leading to $\frac{l^2}{2} I_0 \mu_0 H_0$ for the maximum torque.

Next Steps:

1. Check if the lecture slides mention that the loop's symmetry leads to the final form of $I_0 l^2 \mu_0 H_0$, omitting the factor of $\frac{1}{2}$.
2. Review the problem's assumptions and whether the loop's geometry or number of turns affects this calculation.

Would you like a more detailed breakdown of the relationship between the geometry and torque?

Relative Questions:

1. Why is the magnetic field $\mathbf{B}$ proportional to $\mu_0 \mathbf{H}$?
2. How does changing the side length $l$ of the loop affect the torque?
3. What is the physical meaning of the angle $\theta$ in the torque equation?
4. How would the torque expression change for a non-square rectangular loop?
5. What role does $\mu_0$ (permeability of free space) play in calculating electromagnetic forces?

Tip: When calculating torque, always be mindful of how the radius or lever arm is defined, especially in non-circular geometries.