Question 2: Magnetism and Matter

Question

A long cylindrical conductor with large cross section carries an electric current distributed uniformly over its cross-section. Magnetic field due to this current is:A. maximum at either ends of the conductor and minimum at the midpointB. maximum at the axis of the conductorC. minimum at the surface of the conductorD. minimum at the axis of the conductorE. same at all points in the cross-section of the conductorChoose the correct answer from the options given below:

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

B, C Only

Answer

E Only

Answer

A, D Only

Answer

D Only

Magnetism and Matter

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