Calculating Magnetic Induction at the Center of a Circular Coil

The question asks us to find the new magnetic induction at the center of a circular coil when its physical parameters are changed. We are given the initial magnetic induction and the changes in the number of turns and radius.

Understanding the Formula for Magnetic Field

The magnetic induction (or magnetic field strength) at the center of a circular coil carrying a steady current is given by the formula:

\( B = \frac{\mu_0 N I}{2R} \)

Where:

• \( B \) is the magnetic induction at the center.
• \( \mu_0 \) is the permeability of free space (a constant).
• \( N \) is the number of turns in the coil.
• \( I \) is the steady current flowing through the coil.
• \( R \) is the radius of the coil.

From this formula, we can see that the magnetic field \( B \) is directly proportional to the number of turns \( N \) and the current \( I \), and inversely proportional to the radius \( R \).

Initial Conditions

We are given the initial conditions for the circular coil:

• Initial radius = \( R \)
• Initial number of turns = \( N \)
• Initial current = \( I \) (steady current)
• Initial magnetic induction at the center = \( B = 0.1 \) tesla

Using the formula, the initial magnetic induction can be written as:

\( 0.1 = \frac{\mu_0 N I}{2R} \)

Let's call this Equation (1).

Changed Conditions

The question states that the coil is changed as follows:

• The number of turns is doubled. New number of turns, \( N' = 2N \).
• The radius is halved. New radius, \( R' = \frac{R}{2} \).
• The current remains steady. New current, \( I' = I \).

Calculating the New Magnetic Induction

Now, let's use the formula for magnetic induction with the new parameters \( N' \), \( I' \), and \( R' \) to find the new magnetic induction \( B' \) at the center of the coil:

\( B' = \frac{\mu_0 N' I'}{2R'} \)

Substitute the new values: \( N' = 2N \), \( I' = I \), and \( R' = \frac{R}{2} \):

\( B' = \frac{\mu_0 (2N) (I)}{2(\frac{R}{2})} \)

Let's simplify the expression:

\( B' = \frac{\mu_0 2N I}{R} \)

We can rearrange this expression to relate it to the initial magnetic induction formula:

\( B' = 2 \times \frac{\mu_0 N I}{R} \)

To make it look exactly like Equation (1), we can rewrite it as:

\( B' = 4 \times \frac{\mu_0 N I}{2R} \)

Notice that the term \( \frac{\mu_0 N I}{2R} \) is exactly the initial magnetic induction \( B \) from Equation (1). So, we can substitute \( 0.1 \) tesla for this term:

\( B' = 4 \times (0.1 \text{ T}) \)

\( B' = 0.4 \text{ T} \)

Conclusion

When the number of turns in the circular coil is doubled and the radius is halved, while keeping the current steady, the magnetic induction at the center of the coil becomes 0.4 tesla.

Summary of Changes and Effect

Since \( B \propto N/R \), the combined effect is that the new magnetic field \( B' \) will be proportional to \( (2N) / (R/2) \), which simplifies to \( 4N/R \). This is 4 times the original proportionality \( N/R \). Therefore, the new magnetic field is 4 times the original magnetic field.

\( B_{new} = 4 \times B_{original} \)

\( B_{new} = 4 \times 0.1 \text{ T} = 0.4 \text{ T} \)

Revision Table: Circular Coil Magnetic Field

Additional Information: Magnetic Fields

Magnetic fields are regions around magnets or electric currents where magnetic forces can be observed. Magnetic induction (B) is a vector quantity that describes the strength and direction of the magnetic field at a point.

• Source of Magnetic Fields: Moving electric charges (currents) and intrinsic magnetic moments of elementary particles (like electrons) create magnetic fields.
• Units: The SI unit for magnetic induction is the tesla (T). Another common unit is the gauss (G), where 1 T = 10,000 G.
• Direction of Magnetic Field: For a circular coil, the direction of the magnetic field at the center can be determined using the right-hand rule. If you curl the fingers of your right hand in the direction of the current in the coil, your thumb points in the direction of the magnetic field at the center.
• Applications: Circular coils are fundamental components in many electrical devices, including electromagnets, motors, generators, and scientific instruments like solenoids and Helmholtz coils (which produce uniform magnetic fields).