MCQ on Magnetic Field due to a Thick Current Carrying Conductor

Usually you will be concerned about the magnetic field outside a current carrying conductor. You will usually take for granted that the conductor is thin and practically there is no need to explore within the conductor, even if it is thick. But for any body interested in physics, every thing deserves to be explored! Further, you should be prepared to answer a question of the type given below:

A very long, thick, cylindrical conductor of radius ‘a’ carries a direct current ‘I’. The magnetic field produced at a point P by this current is plotted against the distance of the point P from the centre of the conductor. Which one of the following graphs gives the correct variation?

The magnetic field at any point outside a long conductor carrying a current I, as you may be remembering, is given by

B = μ₀I/2πr, where μ₀ is the permeability of free space and ‘r’ is the distance of the point from the centre of the conductor. So, the field outside is inversely proportional to the distance ‘r’.

The magnetic field inside the conductor can be found using the above expression itself by substituting for ‘I’ as the current carried by the cylindrical portion of the conductor having radius ‘r’.

Therefore, in place of ‘I’ you have to substitute (I/πa²)(πr²) = Ir²/a².

Thus, the field inside, B = μ₀(Ir²/a²)/2πr = μ₀Ir/2πa².

Therefore, the field inside is directly proportional to the distance ‘r’

The correct option is graph (A).

[The magnetic field inside or outside the conductor can be easily found using Ampere’s circuital law: Outside the conductor, since the current enclosed by the closed path of radius ‘r’ is the entire current ‘I’, you will write the law as B×2πr = μ₀I and will obtain B = μ₀I/2πr.

Inside the conductor, since the current enclosed by the closed path of radius ‘r’ is Ir²/a², you will write the law as B×2πr = μ₀(Ir²/a²) and will obtain B = μ₀Ir/2πa²].