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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 attached.

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 [𝐼0*l^2*𝑚0*μ0*𝐻0] [Nm].
Be sure to show a picture of the problem, and provide a numerical result being sure to account for the units.
In order for this to act as a motor, what needs to happen to the current as the loop rotates?

To solve the problem, we need to calculate the maximum torque experienced by a square current loop immersed in a magnetic field. Here's the general approach to solving the problem step by step:

### Key variables and parameters:
- Side length of the square loop: \( l = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Current in the loop: \( I_0 = 600 \, \text{mA} = 0.6 \, \text{A} \)
- Magnetic field strength: \( |H_0| = 5 \, \text{kA/m} = 5000 \, \text{A/m} \)
- \( \mu_0 \) is the permeability of free space: \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)
- \( \mu_0 \) is also used in Ampere’s law to link the magnetic field to the magnetic induction \( B = \mu_0 H_0 \).

### Formula for the torque on a current loop:
The maximum torque \( \tau_{\text{max}} \) on a square loop is given by the formula:
\[
\tau_{\text{max}} = I_0 \cdot A \cdot B
\]
where:
- \( I_0 \) is the current in the loop,
- \( A = l^2 \) is the area of the square loop,
- \( B = \mu_0 \cdot H_0 \) is the magnetic field induction.

Thus, we can express the torque as:
\[
\tau_{\text{max}} = I_0 \cdot l^2 \cdot \mu_0 \cdot H_0
\]

### Step 1: Calculate the magnetic induction \( B \)
\[
B = \mu_0 \cdot H_0 = 4\pi \times 10^{-7} \, \text{T m/A} \cdot 5000 \, \text{A/m}
\]

### Step 2: Compute the area of the square loop
\[
A = l^2 = (0.05 \, \text{m})^2 = 0.0025 \, \text{m}^2
\]

### Step 3: Compute the maximum torque
Now, let's compute the maximum torque using all the values. I will calculate the torque numerically.The maximum torque experienced by the square loop is approximately \( 9.42 \times 10^{-6} \, \text{Nm} \).

### To summarize:
- The maximum torque on the loop is \( 9.42 \times 10^{-6} \, \text{Nm} \).
  
### What needs to happen for this to act as a motor?
In order for the loop to continuously rotate and act as a motor, the direction of the current in the loop needs to reverse every half-turn. This switching of current is typically accomplished using a commutator in a DC motor. It ensures that the torque always acts in the same direction, sustaining the rotational motion.

Would you like further details on any part of the solution, or more clarification on the motor operation?

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Here are 5 related questions:
1. How does the area of the loop affect the torque?
2. What is the role of the commutator in a DC motor?
3. How would the torque change if the loop were rectangular instead of square?
4. How does the direction of the magnetic field affect the torque on the loop?
5. How is torque related to the efficiency of a motor?

**Tip:** The magnetic induction \( B \) depends on both the magnetic field \( H \) and the permeability \( \mu_0 \), so increasing the magnetic field strength will directly increase the torque.

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 attached. 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 |H̅ | = 5[kA/m], the maximum torque experienced by the loop is [𝐼0*l^2*𝑚0*μ0*𝐻0] [Nm]. In order for this to act as a motor, what needs to happen to the current as the loop rotates?

Learn how to calculate the maximum torque experienced by a square current loop in a magnetic field for a DC motor. With a loop side length of 5 cm, current of 600 mA, and a magnetic field strength of 5 kA/m, this example walks through the essential steps to solve for torque using key electromagnetic formulas. Suitable for college-level physics students.

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