The magneitc field produced at the center of a current carrying circular coil of radius r, is

Options

A. directly proportional to r
B. inversely proportional to r
C. directly proportional to `r^(2)`
D. inversely proportional to `r^(2)`

Answer

To find the magnetic field produced at the center of a current-carrying circular coil of radius \( r \), we can follow these steps:

### Step 1: Understand the Biot-Savart Law
The Biot-Savart Law gives us a way to calculate the magnetic field \( dB \) produced by a small segment of current \( I \) flowing through a wire. The law is expressed as:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL \times \hat{R}}{R^2} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( dL \) is the length of the current element,
- \( \hat{R} \) is the unit vector pointing from the current element to the point where the field is being calculated,
- \( R \) is the distance from the current element to the point.

### Step 2: Geometry of the Circular Coil
For a circular coil of radius \( r \), we consider a small element \( dL \) on the coil. The distance \( R \) from this element to the center of the coil is constant and equal to \( r \).

### Step 3: Calculate the Magnetic Field Contribution from Each Element
Since the angle \( \theta \) between \( dL \) and \( \hat{R} \) is \( 90^\circ \) (as \( \hat{R} \) is radial and \( dL \) is tangential), we have:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL}{r^2} \]

### Step 4: Integrate Over the Entire Coil
To find the total magnetic field \( B \) at the center of the coil, we integrate \( dB \) around the entire coil:
\[ B = \int dB = \int \frac{\mu_0}{4\pi} \frac{I \, dL}{r^2} \]
Since \( r \) is constant, we can factor it out of the integral:
\[ B = \frac{\mu_0 I}{4\pi r^2} \int dL \]
The integral \( \int dL \) over the entire circular coil is equal to the circumference of the coil, which is \( 2\pi r \):
\[ B = \frac{\mu_0 I}{4\pi r^2} \cdot 2\pi r \]

### Step 5: Simplify the Expression
Now, simplifying the expression:
\[ B = \frac{\mu_0 I}{4\pi r^2} \cdot 2\pi r = \frac{\mu_0 I}{2r} \]

### Conclusion
Thus, the magnetic field at the center of a current-carrying circular coil of radius \( r \) is given by:
\[ B = \frac{\mu_0 I}{2r} \]
This shows that the magnetic field is inversely proportional to the radius \( r \).