To find the magnitude of the magnetic force acting on a circular loop placed in a magnetic field, we can use the principles of electromagnetism. In this scenario, we have a magnetic field described by \( \mathbf{B} = B_0 \, \mathbf{e}_r \), where \( \mathbf{e}_r \) is the radial unit vector. The loop is positioned in the X-Y plane at a height \( d \) along the Z-axis. Let's break down the solution step by step.
Understanding the Magnetic Field and Loop Configuration
The magnetic field \( \mathbf{B} \) varies with the radial distance from the origin, and it points radially outward. Since the loop is in the X-Y plane, we can visualize it as a circle centered at (0, 0, d) with radius \( a \). The current \( I \) flows around this loop.
Magnetic Force on a Current-Carrying Loop
The magnetic force \( \mathbf{F} \) on a current-carrying conductor in a magnetic field is given by the equation:
- Force Equation: \( \mathbf{F} = I \int d\mathbf{l} \times \mathbf{B} \)
Here, \( d\mathbf{l} \) is the differential length element of the loop, and \( \mathbf{B} \) is the magnetic field at that point.
Parameterizing the Loop
To analyze the loop, we can parameterize it using polar coordinates. The position of a point on the loop can be expressed as:
- Position Vector: \( \mathbf{r}(\theta) = (a \cos \theta, a \sin \theta, d) \)
where \( \theta \) varies from 0 to \( 2\pi \). The differential length element \( d\mathbf{l} \) can be derived as:
- Differential Length: \( d\mathbf{l} = (a \, d\theta \, (-\sin \theta), a \, d\theta \, \cos \theta, 0) \)
Calculating the Magnetic Field at the Loop
Next, we need to evaluate the magnetic field \( \mathbf{B} \) at the location of the loop. The radial distance from the origin to a point on the loop is:
- Radial Distance: \( r = \sqrt{(a \cos \theta)^2 + (a \sin \theta)^2 + d^2} = \sqrt{a^2 + d^2} \)
Thus, the magnetic field at the loop is:
- Magnetic Field: \( \mathbf{B} = B_0 \, \frac{(a \cos \theta, a \sin \theta, d)}{\sqrt{a^2 + d^2}} \)
Cross Product Calculation
Now, we compute the cross product \( d\mathbf{l} \times \mathbf{B} \). This involves calculating:
- Cross Product: \( d\mathbf{l} \times \mathbf{B} \)
After performing the cross product, we will integrate over \( \theta \) from 0 to \( 2\pi \) to find the total force.
Integrating to Find the Total Force
The total force \( \mathbf{F} \) can be expressed as:
- Total Force: \( \mathbf{F} = I \int_0^{2\pi} d\mathbf{l} \times \mathbf{B} \)
After evaluating the integral, we will find that the force has both magnitude and direction. The magnitude of the force can be simplified to:
- Magnitude of Force: \( F = I \cdot B_0 \cdot a \cdot 2\pi \cdot \frac{a}{\sqrt{a^2 + d^2}} \)
Final Result
Thus, the magnitude of the magnetic force acting on the circular loop is:
- Final Expression: \( F = \frac{2\pi I a B_0 a}{\sqrt{a^2 + d^2}} \)
This result shows how the force depends on the current, the radius of the loop, and the characteristics of the magnetic field. The integration process captures the cumulative effect of the magnetic field acting on each segment of the loop, leading to the total force experienced by the loop.
