Question

A current $i$ ampere flows along an infinitely long straight thin-walled tube, then the magnetic induction at any point inside the tube is

• A. Infinite
• B. Zero
• C. $\frac{{\mu }_o}{4\pi }\times \frac{2i}{r}Tesla$
• D. $\frac{2i}{r}Tesla$

Answer 1

The correct option is B

Zero

Explanation for the correct option

B: Zero

Step 1: Given data

Current flowing through the tube$=i$

Step 2: Assumptions

Internal radius of the tube$=R$

Magnetic field density$=B$

Magnetic permeability$={\mu }_o$

Step 3: Formula used

$\oint \overrightarrow{B}.\overrightarrow{dl}={\mu }_{o}I$ …………………(a)

The above equation is the mathematical form of “ampere's Circuital Law”.

In this equation, ($dl$) is the small length element and ($I$) is the current flowing through a loop.

Step 4: Calculation of the magnetic induction at a point inside the tube

Let's draw a hypothetical loop of radius ($r$) from the centre of the tube as shown in the diagram above.

Electric current ($i$) flows only through the tube as shown in the figure.

There is no electric current flowing through the hypothetical loop drawn inside the given cylinder as is evident from the diagram above.

At the centre of the loop, the magnitude of the electric current is also zero.

$I=0$ ………………..(b)

Substituting equation (b) in equation (a), we get

$\oint \overrightarrow{B}.\overrightarrow{dl}=0$

Magnetic induction is zero for all points within the tube provided the point does not lie on the body of the tube.

Hence, option (B) is correct.