QUESTION A DC motor operates by driving a current through a loop that is immersed in an external magnetic field. The interaction between the current and the external field creates a torque on the loop that causes it to rotate. A very simplified model of a DC motor is shown below.

Show that if the loop is a square of side length l = 5 [cm], the current is 𝐼0 = 600.0 [mA], and the external magnetic field strength is |H ̅ | = 5[kA/m], the maximum torque experienced by the loop is [𝐼0l^2μ0𝐻0] [Nm].

MY ANSWER: As shown in Lecture 6 Slide 3, the magnetic field H ̅, and magnetic flux density B ̅, are related as follows, where μ_0 is the permittivity of free space. B ̅=μ_0 H ̅ As shown in Lecture 5 Slide 8, in the presence of an external magnetic field, H ̅, a force is generated. If the section of wire where the force to be calculated is straight, and B ̅ is uniform then, as seen on this online source, we can define the force as F ⃗=I_0 l ⃗×B ̅ F ⃗=I_0 l ⃗×μ_0 H ̅_0 F=I_0 lμ_0 H_0 sin⁡(θ) Where l is the length of the wire. Similarly, as seen in Lecture 5 Slide 8, the torque (i.e. the turning moment of a force about an axis), is defined as follows τ ⃗=r ⃗×F ⃗ τ ⃗=r ⃗×(I_0 l ⃗×μ_0 H ̅_0 ) τ ⃗=r ⃗×(I_0 lμ_0 H_0 sin⁡(θ) ) τ=〖rI〗_0 lμ_0 H_0 sin^2⁡(θ)

Where r is the radius of the object being rotated.

MY QUESTIONS FOR YOU are there any mistakes in my answer so far? I am confused about how l and r related? have I defined these correctly? because I need them to actually be equal in order to get the equation in need to reach (shown in the question stem)