Question 13: Magnetism and Matter

Question

Two identical long current carrying wires are bent into the shapes shown in the following figures. If the magnitude of magnetic fields at the centres P and Q of a semicircular arc are $B_1$ and $B_2$ respectively, then the ratio $\frac{B_1}{B_2}$ is ______.

Accepted Answer

Consider a long, straight wire of radius R carrying a steady current I. Since the current distribution is uniform, the current density (J) is constant throughout the cross-section.
The current density J is given by :
$$ \mathrm{J}=\frac{\mathrm{I}}{	ext { Total Area }}=\frac{\mathrm{I}}{\pi \mathrm{R}^2} $$
AmpÃÂ¨re's Law states that the line integral of the magnetic field B around a closed loop (Amperian loop) is equal to $\mu_0$ times the current enclosed ( $\mathrm{I}_{\mathrm{enc}}$ ) by that loop :
$$ \oint \mathrm{B} \cdot \mathrm{dl}=\mu_0 \mathrm{I}_{\mathrm{enc}} $$
To find the field inside, consider a circular Amperian loop of radius $r$ (where $r \mathrm{R})$ :
Since the Amperian loop (radius $r$ ) completely surrounds the entire cross-section of the cylinder (radius R), it encloses the total current I flowing through the conductor.
$$ \mathrm{I}_{\mathrm{enc}}=\mathrm{I} $$
Using AmpÃÂ¨re's Law equation :
$$ \mathrm{B}(2 \pi \mathrm{r})=\mu_0 \mathrm{I} $$
$$\Rightarrow $$ $$ B=\frac{\mu_0 I}{2 \pi r} $$
This shows that B is inversely proportional to $\mathrm{r}\left(\mathrm{B} \propto \frac{1}{\mathrm{r}}ight)$.
The field is maximum at the surface ( $\mathrm{r}=\mathrm{R}$ ) and decreases when moving further away. Hence, statement C is incorrect.
The field distribution depends on the distance from the axis, not the length/midpoint of the conductor. Hence, statement (A) is incorrect.
Since, the field varies linearly inside and inversely outside; it is not uniform. Hence, option E is incorrect.
Therefore, only statement D is correct.
Hence, the correct option is (D).

Answer

$\frac{2 + \pi}{1 + \pi}$

Answer

$\frac{1 + \pi}{1 - \pi}$

Answer

$\frac{2 + \pi}{1 - \pi}$

Answer

$\frac{1 + \pi}{2 - \pi}$

Magnetism and Matter

Two identical long current carrying wires are bent into the shapes shown in the following figures. If the magnitude of magnetic fields at the centres P and Q of a semicircular arc are B_1 and B_2 respectively, then the ratio \frac{B_1}{B_2} is ______.

Choose the correct answer: