Consider a torod , having a circular cross-section of radius b, major radius R ( R > > b), having N turns and carrying current I. Find the total energy stored in the toroid.

Options

A. `(mu_0N^2I^2b^2)/(2R)`
B. `(mu_0N^2I^2b^2)/(3R)`
C. `(mu_0N^2I^2b^2)/(6R)`
D. `(mu_0N^2I^2b^2)/(4R)`

Answer

To find the total energy stored in a toroid with a circular cross-section, we can follow these steps:

### Step 1: Understand the Geometry of the Toroid
A toroid has a circular cross-section with a radius \( b \) and a major radius \( R \). Given that \( R \gg b \), we can treat the toroid as a solenoid wrapped around a circular path.

### Step 2: Determine the Magnetic Field Inside the Toroid
The magnetic field \( B \) inside a toroid is given by the formula:
\[ B = \frac{\mu_0 N I}{2 \pi R} \]
where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, and \( I \) is the current flowing through the toroid.

### Step 3: Calculate the Area of the Cross-Section
The area \( A \) of the circular cross-section of the toroid is:
\[ A = \pi b^2 \]

### Step 4: Calculate the Magnetic Flux
The magnetic flux \( \Phi \) through one turn of the toroid is given by:
\[ \Phi = B \cdot A = \left(\frac{\mu_0 N I}{2 \pi R}\right) \cdot (\pi b^2) = \frac{\mu_0 N I b^2}{2 R} \]
Since there are \( N \) turns, the total magnetic flux linked with the toroid is:
\[ \Phi_{\text{total}} = N \Phi = N \cdot \frac{\mu_0 N I b^2}{2 R} = \frac{\mu_0 N^2 I b^2}{2 R} \]

### Step 5: Calculate the Inductance of the Toroid
The inductance \( L \) of the toroid can be calculated using the formula:
\[ L = \frac{\Phi_{\text{total}}}{I} = \frac{\frac{\mu_0 N^2 I b^2}{2 R}}{I} = \frac{\mu_0 N^2 b^2}{2 R} \]

### Step 6: Calculate the Energy Stored in the Toroid
The energy \( U \) stored in an inductor is given by:
\[ U = \frac{1}{2} L I^2 \]
Substituting the expression for \( L \):
\[ U = \frac{1}{2} \left(\frac{\mu_0 N^2 b^2}{2 R}\right) I^2 = \frac{\mu_0 N^2 b^2 I^2}{4 R} \]

### Final Result
The total energy stored in the toroid is:
\[ U = \frac{\mu_0 N^2 b^2 I^2}{4 R} \]

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