CBSE Class 12 · 2013 · Magnetic Effects of CurrentLong Answer

WhatsApp Email

(a) Using Biot-Savart's law, derive the expression for the magnetic field in vector form at a point on the axis of a circular current loop.
(b) What does a toroid consist of? Find the expression for the magnetic field inside a toroid of

N

turns, average radius

r

, carrying a current

I

. Show that the magnetic field in the open space inside (exterior to) the toroid is zero.

Answer: (a) $B = \frac{μ _{0} I R ^{2}}{2 ( R ^{2} + x ^{2} ) ^{3/2}}$ .
(b) $B = \frac{μ _{0} N I}{2 π r}$ inside; zero outside.

Solution

(a) By Biot-Savart each element gives

d B = \frac{μ _{0} I d ℓ}{4 π ( R ^{2} + x ^{2} )}

; perpendicular parts cancel and axial parts add to

B = \frac{μ _{0} I R ^{2}}{2 ( R ^{2} + x ^{2} ) ^{3/2}} (along the axis) .

(b) A toroid is a solenoid bent into a closed ring. Take a circular amperian loop of radius

r

inside the core:

\oint B \cdot d ℓ = B (2 π r) = μ_{0} N I

, so

B = \frac{μ _{0} N I}{2 π r} .

For a loop in the open space inside the ring (the hole), no current is enclosed, and for a loop outside the toroid the net enclosed current is also zero; hence

B = 0

in both exterior regions.

CBSE Class 12 Board · 2013 · Subjective

Practise more: all CBSE board PYQs on Magnetic Effects of Current →

CBSE Class 12 · 2013 · Magnetic Effects of CurrentLong Answer

Solution

More board questions on Magnetic Effects of Current