Question

A toroid with thick windings of $N$ turns has inner and outer radii $R_{1}$ and $R_{2}$ respectively. If it carries certain steady current I, the variation of the magnetic field due to the toroid with radial distance is correctly graphed in

• A.

  

• B.

  

• C.

  

• D.

  

Answer 1

The Correct Option is D

This question requires us to understand the magnetic field distribution in a toroid with thick windings. Let's analyze the problem step-by-step.

A toroid is a coil shaped like a donut, and its magnetic field properties are unique. The inner radius of the toroid is \( R_1 \), and the outer radius is \( R_2 \). The number of turns is \( N \), and it carries a steady current \( I \). 

The magnetic field inside the toroid varies with the radial distance from the center. We use Ampère's Law to deduce this:

\[\oint \vec{B} \cdot d\vec{l} = \mu_0 N I\]

In simpler terms, inside a toroid, the magnetic field \( B \) is directly proportional to current \( I \) and inversely proportional to the radial distance \( r \), particularly between the inner radius \( R_1 \) and the outer radius \( R_2 \). The magnetic field is zero outside this range (\( r < R_1 \) or \( r > R_2 \)) since the windings do not exist there to enclose the magnetic field.

Therefore, the relationship of the magnetic field with respect to radial distance \( r \) is as follows:

• For \( r < R_1 \), \(B = 0\)
• For \( R_1 \leq r \leq R_2 \), \(B \propto \frac{1}{r}\)
• For \( r > R_2 \), \(B = 0\)

The graph that correctly represents this variation will show zero field from the center to \( R_1 \), an inversely proportional decrease between \( R_1 \) and \( R_2 \), and again zero beyond \( R_2 \).

Therefore, the correct graph should exhibit a region of zero magnetic field for \( r < R_1 \), a declining section between \( R_1 \) and \( R_2 \), and again zero beyond \( R_2 \). The last image matches this description accurately.

Thus, the correct variation of the magnetic field with radial distance is shown by the image above.

Answer 2

$B =\mu_{0} nI$ where $n =\frac{ N }{2 \pi r }$ $B=\frac{\mu_{0} N I}{2 \pi\left(\frac{R_{1}+R_{2}}{2}\right)} $ $B=\frac{\mu_{0} N I}{\pi\left(R_{1}+R_{2}\right)}$