An ideal toroidal solenoid (see Example 28.10 ) has inner radius r1=13.8 cm and outer radius r2

step 1 Hi, here in this given problem for the ideal toroidal solenoid. Its inner radius r1, that is 13

An ideal toroidal solenoid (see Example 28.10 ) has inner...

An ideal toroidal solenoid (see Example 28.10 ) has inner radius $r_{1}=13.8 \mathrm{~cm}$ and outer radius $r_{2}=17.3 \mathrm{~cm} .$ The solenoid has 300 turns and carries a current of $8.20 \mathrm{~A}$. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) $10.8 \mathrm{~cm}:$ (b) $16.5 \mathrm{~cm}:$ (c) $20.0 \mathrm{~cm} ?$

An ideal toroidal solenoid (see Example 28.10 ) has inner radius $r_{1}=13.8 \mathrm{~cm}$ and outer radius $r_{2}=17.3 \mathrm{~cm} .$ The solenoid has 300 turns and carries a current of $8.20 \mathrm{~A}$. What is the magnitude of the magnetic field at the following distances from the center of the torus:
(a) $10.8 \mathrm{~cm}:$
(b) $16.5 \mathrm{~cm}:$
(c) $20.0 \mathrm{~cm} ?$

Key Concepts

Ampere's Law

Ampere's Law relates the magnetic field circulating around a closed loop to the total electric current passing through the loop. In the context of a toroidal solenoid, this law is applied by selecting an appropriate circular path inside the torus, which allows for a straightforward calculation of the magnetic field magnitude by equating the line integral of the magnetic field to ?? times the enclosed current.

Magnetic Field Distribution in Toroids

In an ideal toroidal solenoid, the magnetic field tends to follow a 1/r dependence inside the core, where r is the radial distance from the center of the torus, due to the geometry and winding structure. This distribution means that the magnetic field can vary with distance from the center, being well-defined within the region of the windings while being nearly zero outside this region. Understanding this helps in computing the field at various radial positions relative to the torus.

Toroidal Solenoid

A toroidal solenoid is a coil of wire that is wound into a torus (doughnut) shape. In this configuration, the magnetic field is mostly confined within the core of the torus, resulting in minimal external magnetic fields. This makes the toroid an ideal case for applying symmetry arguments when determining the behavior of the magnetic field using theoretical principles.

Transcript

00:01 Hi, here in this given problem for the ideal toroidal solenoid.

00:17 Its inner radius r1, that is 13 .8 centimeter, and its outer radius r2, that is 17 .3 centimeters so average radius of its core that will be r is equal to r1 plus r2 divided by 2 means this is 13 .8 plus 17 .3 divided by 2 and it is calculated to be equal to 15 .5 5 cm total turns in this toroid these are 3 current passing through the toroid 8 .20 ampere.

01:16 In the first part of the problem distance of observation point that is 10 .8 centimeter means this is within the empty space of solenoid because if we look at a solenoid, it is a structure obtained by winding large number of turns.

01:52 The conducting wire over a ring that is known as anchor ring.

01:59 So this is the radius which we have found and in the first case observation point is here in the empty region and at that point as there will be no imperial loop sorry as there will be no current threading the imperial loop passing through this point so we conclude magnetic field here that will be zero.

02:23 There will be no magnetic field passing through it.

02:26 Answer for the first part of the problem then in the second part of the problem this time r is given as 16 .5 centimeter means the point is somewhere within the solenoid and there are currents passing through this loop passing through this ampereal loop now.

02:49 So b will be given by mu not n i and is the number of turns per unit length means n i by 2 pi r so plugging in the known values for mu not this is or we can multiply it by 2 by 2 making it mu not 2 n i divided by 4 pi divided by r so this mu not upon 4 pi that is 10 x x x x by 2 times of 300 multiplied by current 8 .20 and divided by the distance and sorry that is a radius and this radius average radius of the solenoid this is 2 times here this is 2 sorry this is not 2 over here this is just r which is 15.

03:55 0 .55 multiplied by 10 to the power minus 2...

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