Magnetism Concept Page - 10

Example

Magnetic field due to a toroid

The magnetic field in the open space inside (point P) and exterior to the toroid (point Q) is zero. The field B$B$ inside the toroid is constant in magnitude for the ideal toroid of closely wound turns.The direction of the magnetic field inside is clockwise as per the right-hand thumb rule for circular loops. Three circular Amperian loops 1, 2 and 3 are shown by dashed lines. By symmetry, the magnetic field should be tangential to each of them and constant in magnitude for a given loop.Example:The number of turns per unit length in a toroid is 103${10}^3$ and current flowing in it is 14π${\displaystyle \frac{1}{4\pi }}$ ampere, then the magnetic induction produced in it, is :Magnetic field in a toroid B=μ0ni$B={\mu }_{0}ni$
Given n=103$n={10}^3$ and i=14πA$i={\displaystyle \frac{1}{4\pi }}A$
We know that μ0=4π×10−7T/A${\mu }_0=4\pi \times {10}^{-7}T/A$
B=4π×10−7×103×14π=10−4T$B=4\pi \times {10}^{-7}\times {10}^3\times {\displaystyle \frac{1}{4\pi }}={10}^{-4}T$

Formula

Number of turns in a toroidal coil

The figure below shows a cross-sectional view of the inner radius of a toroid inductor and wire. The inner radius of the torus is A, the radius of the wire is r, and the maximum number of loops is n.The equation that relates A, r, and n is:sin(πn)=rA−r$sin({\displaystyle \frac{\pi }{n}})={\displaystyle \frac{r}{A-r}}$ in radians

n=πarcsinrA−r$n={\displaystyle \frac{\pi }{\arcsin {\displaystyle \frac{r}{A-r}}}}$

Definition

Observe that parallel currents repel in a solenoid and toroid

Two parallel wires carrying currents will either attract or repel each other.
Consider diagram (a):
Apply the right-hand grip rule to the left-hand conductor - this indicates that the magnetic field at the right-hand conductor due to the current in the left-hand conductor is into the paper.
Now apply Flemings left hand rule to the right-hand conductor - this indicates that the field produces a force on the right-hand conductor to the left, as shown.
The directions of all the forces can be determined in a similar way.
The flux density B1$B_1$ produced by the left-hand conductor at the right-hand conductor is given by: B1=μ0I12πr$B_1={\displaystyle \frac{{\mu }_0{I}_1}{2\pi r}}$

Definition

Strength of electromagnet

The strength of the magnetic field produced by an electromagnet depends on the number of turns in the coil, the magnitude of the current and the magnetic permeability of the core material.

Definition

Cyclotron

The cyclotron is a machine to accelerate charged particles or ions to high energies.The cyclotron uses both electric and magnetic fields in combination to increase the energy of charged particles.As the fields are perpendicular to each other they are called crossed fields. Cyclotron uses the fact that the frequency of revolution of the charged particle in a magnetic field is independent of its energy. The particles move most of the time inside two semicircular disc-like metal containers, D1 and D2, which are called dees as they look like the letter D.Inside the metal boxes the particle is shielded and is not acted on by the electric field. The magnetic field, however, acts on the particle and makes it go round in a circular path inside a dee. Every time the particle moves from one dee to another it is acted upon by the electric field. The sign of the electric field is changed alternately in tune with the circular motion of the particle.This ensures that the particle is always accelerated by the electric field.Each time the acceleration increases the energy of the particle. As energy increases, the radius of the circular path increases. So the path is a spiral one.

Definition

Cyclotron frequency

The cyclotron frequency or gyrofrequency is the frequency of a charged particle moving perpendicular to the direction of a uniform magnetic field B$B$ (constant magnitude and direction).
vc=qB2πm$v_c={\displaystyle \frac{qB}{2\pi m}}$
T=1vc$T={\displaystyle \frac{1}{v_c}}$
where m$m$ is the particle mass, q$q$ its charge and B$B$ the magnetic field, T$T$ is the time period and vc$v_c$ is the cyclotron frequency.

Definition

Uses of cyclotron

The cyclotron is used to bombard nuclei with energetic particles, so accelerated by it, and study the resulting nuclear reactions. It is also used to implant ions into solids and modify their properties or even synthesise new materials. It is used in hospitals to produce radioactive substances which can be used in diagnosis and treatment.

Definition

Describe the limitations of cyclotron

Limitations of Cyclotron are as follows:

• Cyclotron cannot accelerate uncharged particles like neutrons.
• Cyclotron cannot accelerate electrons because of its small mass.
• It cannot accelerate positively charged particles with large mass due to relativistic effect.

Formula

Force between two parallel current carrying wires.

There are two long wires a$a$ and b$b$ carrying currents Ia$I_a$ and Ib$I_b$ respectively, separated by a distance d$d$, then the force on a segment of length L$L$ of b$b$ due to a$a$ is
Fba=IbLBa=μ0IaIb2πdL$F_{ba}=I_{b}L{B}_a={\displaystyle \frac{{\mu }_0{I}_a{I}_b}{2\pi d}}L$

Example

Force between parallel current carrying sheets

A system consists of two parallel planes carrying currents producing a uniform magnetic field of induction B$B$ between the planes. Outside this space there is no magnetic field. If the magnetic force acting per unit area of each plane is F1=B2xμ0${\displaystyle F_1=\frac{B^2}{x{\mu }_0}}$. Find x$x$By the circulation theorem B=μ0i$B={\mu }_{0}i$,
where i=$i=$ current per unit length flowing along the plane perpendicular to the paper. currents flow in the opposite sense in the two planes and produce the given field B$B$ by superposition.
The field due to one of the plates is just 12B${\displaystyle \frac{1}{2}B}$. The force on the plate is,
12B×i×Length×Breadth=B22μ0${\displaystyle \frac{1}{2}B\times i\times Length\times Breadth=\frac{B^2}{2{\mu }_0}}$ per unit area.
(Recall the formula F=BIl$F=BIl$ on a straight wire)

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