A toroidal solenoid with 400 turns of wire and a mean radius of 6.0 cm carries a current of 0.2

A toroidal solenoid with 400 turns of wire and a mean radius...

A toroidal solenoid with 400 turns of wire and a mean radius of $6.0 \mathrm{~cm}$ carries a current of 0.25 A. The relative permeability of the core is $80 .$ (a) What is the magnetic field in the core? (b) What part of the magnetic field is due to the magnetic moments of the atoms in the core?

Key Concepts

Ampere’s Law

Ampere’s Law is a fundamental principle in electromagnetism that relates the integrated magnetic field around a closed loop to the net current passing through that loop. It is especially useful in calculating the magnetic field in systems with a high degree of symmetry such as solenoids and toroids.

Toroidal Geometry

A toroidal solenoid is a coil wound in the shape of a torus (doughnut), which confines the magnetic field primarily within its core. Due to its circular symmetry, Ampere’s law can be applied along a circular path following the toroid's mean radius to compute the magnetic field.

Magnetic Permeability

Magnetic permeability measures how a material responds to an applied magnetic field, defining the ease with which magnetic lines of force can pass through it. It is given by the product of the vacuum permeability (??) and the relative permeability (?r), and it plays a crucial role in determining the strength of the magnetic field inside magnetic materials.

Magnetization

Magnetization refers to the magnetic moment per unit volume that results from the alignment of atomic magnetic moments within a material. In the context of a magnetic core, the overall magnetic field is the sum of the contribution generated by the free currents (described by the H-field) and the additional contribution due to the induced magnetization (M), which is accounted for in the relation B = ?? (H + M).

Transcript

00:01 High in the given problem number of turns in the toroidal solenoid is given as n is equal to 400.

00:21 Mean radius of this solenoid is toroidal solenoid is r is equal to 6 .0 centimeter or we can say this is 6 .0 into 10 dash per minus 2 meter.

00:40 The current carried into this toroidal solenoid is 0 .25 ampere and relative permeability of the core material within this solenoidase m u .r is equal to 80.

00:57 Now we know the relative permeability is given as the absolute permeability of the core material divided by the absolute permeability of the free space.

01:07 So this absolute permeability of the core material will be given by the product of its relative permeability with the absolute permeability of the free space.

01:19 Hence, in the first part of the problem, the expression for the magnetic field within the core of this solenoid, the dueroidal solenoid will be given by the absolute permeability of the core material multiplied by number of turns per unit length multiplied by the current carried by this solenoid.

01:39 Now for mu, it will become mu r into mu not.

01:43 For number of turns per unit length, this is number of turns, capital n, divided by the length of the toroid, which is nothing but the circumference of the toroid and multiplied by the current carried by it.

01:56 So, plugging in all known values, this b will be given by for mu r, this is 80.

02:06 For mu not, this is, 4 pi into 10 dash par minus 7 tesla meter per ampere.

02:13 For number of turns, this is 400.

02:16 The current carried by it 0 .25 ampere and then divided by 2 pi into r.

02:29 So 2 pi as it is for the radius...

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