Question

Which of the following conclusions can be drawn from the result \[\oint {\vec B.d\vec A} = 0\] ?
(A) Magnetic field is zero everywhere.
(B) Magnetic monopole cannot exist.
(C) Magnetic lines of force do not intersect each other.
(D) A current produces a magnetic field.

Answer 1

Hint: The dot product of magnetic field and area gives us the magnetic flux. The net magnetic flux of a dipole is always zero.

Complete step by step answer:
In the given question, we are supplied with an equation which is shown below:
\[\oint {\vec B.d\vec A} = 0\]
We are asked to draw the conclusion from the equation, so as which one of the options holds truth for this.
To begin with, let us write the above equation again,
\[\oint {\vec B.d\vec A} = 0\]
Where,
\[\vec B\] indicates the magnetic field vector.
\[d\vec A\] indicates the small area vector.
The above equation can be modified as follows:
If we take the whole area of study, then after integration it will become like this as shown below.
\[\vec B.\vec A = 0\]
We can again, so some modification in there as:
\[BA\cos \theta = 0\]
The above equation gives us the magnetic flux which is referred to as the integral sum of all the magnetic fields which are passing through the infinitesimal area elements. It is an effective tool for explaining the effects of the magnetic force on anything that occupies a given area. Magnetic flux calculation is related to the specific area selected.From the given equation, we can draw some conclusions that the flux of a certain closed surface is shown as zero which tells that the ret or resultant magnetic charge is also equal to zero. This holds good if there are poles with equal magnitude but opposite polarities, then only the magnetic charge can cancel out each other, which gives the result as zero.So, for this situation, we must need two poles which are identical in nature but opposite in polarity.Hence, we can draw a conclusion that there can’t exist monopoles in real life. Monopoles are not feasible.

The correct option is B.

Note: While solving this problem, we should remember that \[\theta \] is the angle formed between a perpendicular vector to the area and the magnetic field. The magnetic field has to pass through two different areas so that their net magnetic charge becomes zero, giving rise to two opposite poles at every instant.