Question

A toroid with thick windings of N turns has inner and outer radii R₁ and R₂ 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

Understanding Magnetic Field Variation in a Toroid:

The magnetic field inside a toroid with thick windings, carrying a steady current $I$, varies with the radial distance ($r$). For a toroid with inner radius $R_1$ and outer radius $R_2$, the magnetic field $B$ at a point inside the windings (at a radial distance $r$ from the center axis, where $R_1 \leq r \leq R_2$) is given by Ampere's Law. Assuming the windings are uniformly distributed, the magnetic field inside the toroid is approximately tangential to circles centered on the toroid's axis and its magnitude is given by:

$B = \frac{\mu_0 N I}{2\pi r}$

where:

- $\mu_0$ is the permeability of free space
- $N$ is the total number of turns
- $I$ is the current
- $r$ is the radial distance from the center of the toroid axis

Analyzing the Magnetic Field Variation with Radial Distance:

1. Inside the Toroid Windings ($R_1 \leq r \leq R_2$):

- The magnetic field $B$ is inversely proportional to the radial distance $r$ ($B \propto \frac{1}{r}$).
- As $r$ increases from $R_1$ to $R_2$, the magnetic field $B$ decreases.
- The magnetic field is non-zero in this region.

2. Inside the Hole of the Toroid ($r < R_1$):

- If we consider an Amperian loop inside the hole of the toroid (radius $r < R_1$), this loop encloses no current.
- By Ampere's Law, $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}} = 0$.
- Therefore, the magnetic field inside the hole of the toroid is zero ($B = 0$ for $r < R_1$).

3. Outside the Toroid ($r > R_2$):

- For an ideal toroid with uniformly wound windings, the magnetic field outside is ideally zero. In a real toroid, it's very weak and often approximated as zero.

Evaluating the Given Graphs:

(A) This graph shows a linear variation inside the toroid which is incorrect; the variation should be $1/r$.

(B) This graph shows a linearly increasing field inside the toroid which is incorrect; it should be decreasing as $r$ increases.

(D) This graph correctly shows:
- Magnetic field is zero for $r < R_1$;
- Magnetic field is non-zero and decreases as $r$ increases from $R_1$ to $R_2$. The curve shape resembles a $1/r$ variation;
- The values at $r = R_1$ and $r = R_2$ are shown as $\frac{\mu_0 NI}{2\pi R_1}$ and $\frac{\mu_0 NI}{2\pi R_2}$, which are consistent with the theoretical formula for the magnetic field at the inner and outer radii.

(C) This graph shows a constant magnetic field between $R_1$ and $R_2$, which is incorrect. The field should vary with $r$.

Conclusion:

Graph (C) correctly represents the variation of the magnetic field due to a toroid with radial distance. It shows zero field inside the hole, a decreasing field within the windings as radial distance increases, and the magnitudes at $r = R_1$ and $r = R_2$ are consistent with the theoretical formula.

Final Answer: The final answer is (D)

Answer 2

A toroid, regardless of winding thickness, has a zero magnetic field outside its structure (both inside the toroidal cavity and outside the outer radius).

The windings themselves create the field, but they are confined within the toroid structure between R1 and R2. 

Therefore, the correct graphical representation shows a constant field (represented by the horizontal line) between R1 and R2, where the windings are, and zero elsewhere. 

The correct answer is indeed (D):