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Toroid - Equations, Practice problems, FAQs

Toroid - Equations, Practice problems, FAQs

Solenoid and toroid make use of coils which generate a magnetic field along their axis when a current is passed through them. The amount of magnetic field produced is directly proportional to the current passed through them. In order to avoid magnetic flux losses, a soft iron core is inserted in the space between the coil windings. Toroids make use of a donut shaped core. Being lighter, and capable of producing more magnetic field than solenoid with the same number of turns, they are used in computers, TV sets and audio players. In this article, we will explore toroid in detail. Just imagine that a ring shaped magnet will be made only by winding coils around the doughnut/ring shaped core.

Table of contents

  • What is a toroid?
  • Magnetic field at different points of a toroid
  • Practice problems
  • FAQs

What is a toroid?

A toroid is a solenoid that is bent in the shape of a ring. The earliest form of a toroid is the Rowland ring. A toroid can be termed as a round coil with many turns.

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Magnetic field at different points of a toroid

Let us consider a toroid with inner radius r1 and outer radius r2.

1) For a point inside the toroid, (r<r1<r2)

Let B be the magnetic field. Then according to Ampere’s circuital law,

B .dl =μ0iinside

The current enclosed by the Amperian loop (denoted by yellow dotted lines) is zero.

iinside=0 B=0.

So, the magnetic field inside the toroid is zero.

2) For a point inside the coil of the toroid, (r1<r<r2)

B.dl =μ0ienclosed=Bdl cos 180o=-Bdl 

-Bl=-B(2πr)=-μ0Ni

B=μ0Ni2πr=μ0ni

Here n=N2πr is the number of turns per unit length.

3) For a point outside the toroid, r>r2.

Applying Ampere’s circuital law,

B .dl =μ0inet

The net current enclosed in this Amperian loop is zero, hence inet=0.

The amount of current going into the area of the Amperian loop is equal to the current coming out. So inet=0.

So, B=0.

Hence, the magnetic field outside the toroid is zero.

Practice problems

Q. A toroid has inner radius 25 cm and outer radius 26 cm with 3500 turns and 11 A current flowing through it. Find the magnetic field in the regions marked (I), (II) and (III).

A. In the regions II and III, since they are outside and inside the toroid respectively,

BII=BIII=0. This is because the net current passing through the area of any Amperan coil in this region is zero.

In the region I, there will be a non-zero magnetic field.

Average radius of the coil, R=25+262=25.5 cm

R=25.5×10-2m.

B=μ0Ni2πR=4π×10-7×3500×112π×25.5×10-2 30 mT.

B=0.03 T

Q. Two toroids have 200 and 100 total number of turns respectively. Their average radii are 40 cm and 20 cm. They carry same current i. The ratio of the magnetic fields along the two loops is,

(a) 1:2 (b) 1:1 (c) 4:1 (d) 2:1

A. b

Given,

N1=200 turns, N2=100 turns

R2=0.2 m, R1=0.4 m

B1=0N1i2R1 and B2=0N2i2R2

B1B2=N1N2×R2R1=200100×0.20.4=1:1

Q. The current flowing through two toroids of same size are i and 2i respectively. If both the coils have the same number of turns, calculate the ratio of magnetic fields in both the coils.

(a) 1:2 (b) 1:1 (c) 4:1 (d) 2:1

A. a

i1=i, i2=2i

For the toroids of same size and with same number of turns, magnetic field B ∝ current(i)

i.e., B1B2=i1i2=i2i=1:2

Q. The magnetic field on a point on the toroid is found to be B=0.002 T. If the average radius of the toroid is 25 cm, and it has 1000 turns, calculate the current flowing through it.

A. Given, B=0.002 T, R=25×10-2m, N=1000 turns

B=0Ni2R=410-7103i22510-2

0.002=210-2i25

i=2.5 A.

FAQs

Q. Why does toroid have no polarity ?
A.
If we consider toroid as a sum of many short magnets connected end to end and forming a circular shape, then at each head of the short magnets there is a tail of the next short magnet. Hence there is no south - north polarity.

Q. What is the principle of a toroid ?
A.
Whenever a current is passed through a coil, then there is a magnetic field produced on the windings. The magnitude of the magnetic field produced is directly proportional to the amount of current passed through the windings and the number of turns of the coils. This is in accordance with Biot Savart’s law.

Q. What is a toroid made of?
A.
It is made up of insulated or enameled wire which is wound on a donut shaped ring. The ring is made of powdered iron and it prevents magnetic losses occurring between the windings.

Q. Is the magnetic field at a point inside the coil of the toroid constant if the current is constant?
A.
Yes, the magnetic field at any point on the surface of the toroid is constant. This is because the current passing through the area of any Amperian loop in this region is also constant.

The magnetic field outside and inside are zero.

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