To solve the problem, we need to calculate the torque (\(\tau\)) on the rectangular coil when it is tilted at an angle of \(45^\circ\) about the Z-axis in a magnetic field. The formula for torque on a current-carrying coil in a magnetic field is given by:
\[
\tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta)
\]
Where:
- \(n\) = number of turns of the coil
- \(I\) = current flowing through the coil
- \(A\) = area of the coil
- \(B\) = magnetic field strength
- \(\theta\) = angle between the normal to the coil and the magnetic field
### Step 1: Calculate the area of the coil
The dimensions of the coil are given as \(5 \, \text{cm} \times 2.5 \, \text{cm}\). First, we convert these dimensions into meters:
\[
5 \, \text{cm} = 0.05 \, \text{m} \quad \text{and} \quad 2.5 \, \text{cm} = 0.025 \, \text{m}
\]
Now, we calculate the area \(A\):
\[
A = \text{length} \times \text{width} = 0.05 \, \text{m} \times 0.025 \, \text{m} = 0.00125 \, \text{m}^2
\]
### Step 2: Identify the other parameters
From the problem statement, we have:
- Number of turns, \(n = 100\)
- Current, \(I = 3 \, \text{A}\)
- Magnetic field, \(B = 1 \, \text{T}\)
- Angle, \(\theta = 45^\circ\)
### Step 3: Calculate the torque
Now we can substitute these values into the torque formula:
\[
\tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta)
\]
Substituting the known values:
\[
\tau = 100 \cdot 3 \cdot 0.00125 \cdot 1 \cdot \sin(45^\circ)
\]
We know that \(\sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707\).
Now substituting this value:
\[
\tau = 100 \cdot 3 \cdot 0.00125 \cdot 1 \cdot 0.707
\]
Calculating step-by-step:
1. \(100 \cdot 3 = 300\)
2. \(300 \cdot 0.00125 = 0.375\)
3. \(0.375 \cdot 0.707 \approx 0.265125\)
Thus, the torque \(\tau \approx 0.265 \, \text{N m}\).
### Step 4: Rounding to appropriate significant figures
Rounding \(0.265 \, \text{N m}\) gives us approximately \(0.27 \, \text{N m}\).
### Final Answer
The torque on the coil when tilted at \(45^\circ\) is:
\[
\boxed{0.27 \, \text{N m}}
\]
