Magnetism is a phenomenon due to which moving charges (or magnets) attract ferromagnetic objects and repel diamagnetic objects.
The magnetic force is a consequence of the electromagnetic force, one of the four fundamental forces of nature, and is caused by the motion of charges. Two objects containing charge with the same direction of motion have a magnetic attraction force between them.
Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field B is defined in terms of force on moving charge in the Lorentz force law.
Lorentz force is defined as the combination of the magnetic and electric force on a point charge due to electromagnetic fields. It is used in electromagnetism and is also known as the electromagnetic force. In the year 1895, Hendrik Lorentz derived the modern formula of Lorentz force.
Lorentz force formula for the charged particle is as follows:
F=q(E+v∗B)
Where,
F is the force acting on the particle
q is the electric charge of the particle
v is the velocity
E is the external electric field
B is the magnetic field
For a charge q moving with velocity v in the presence of magnetic field B, force FB is given by:
FB = q(vxB) = qvB(sinθ)ȓ
When motion of charge v and the magnetic field B are at right angle (90°) to each other, the charge will follow a circular path with force FB always acting towards the center (Centripetal force) and the velocity vacting tangentially to the circle.
When a charge in motion moves such that the angle between the moving charge and the plane of magnetic field is θ, then the velocity (v) of charge has 2 components, one component along the direction of magnetic field (vcosθ), and the another perpendicular to the magnetic field (vsinθ).
When the electric field, the magnetic field , and the motion of charge are mutually perpendicular to each other (as shown in the image) , then they are called as crossed fields, and forces due to electric and magnetic fields will act in the opposite directions. So, the Lorentz force F will be:
F = qEî+(qvî x Bk̂) = qEĵ - qVBî = q(E - vB)ĵ
The cyclotron is a machine to accelerate charged particles or ions to high energies. It was invented by E.O. Lawrence and M.S. Livingston in 1934 to investigate nuclear structure. 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.
According to the Biot-Savart law, magnetic field dBdue to current element idl, at a pointP situated at distancer from the current element idl is directly proportional to the current element idl, ii) directly proportional to the sine of the angle (θ) between current element and r, and iii) inversely proportional to the square of the distance r between current element and the point
dB∝idl (sinθ)/r2
dB = (μo/4π)×idl×(sinθ)/r2
dB = idl × r / r3
Here proportionality constant is μo/4π = 10-7Tm/A, and μois the permeability of free space (vacuum).
According to the Biot-Savart law, magnetic field dB at point P due to current element idl in the above diagram is given by:
∴ B = ∫(μo/4π)idlcosθ /x2 = (μo/4π)∫idlcosθ /x2……………….(i)
dB = (μo/4π)idl sin(90°-θ) /x2 = (μo/4π)idlcosθ /x2
Considering triangle ABN:cosθ = AN/dl
AN = dl cosθ
Considering triangle ANP: sin(dθ)˜dθ = AN/x
AN = x(dθ)
Using the value of ANfrom the above 2 equations:
dlcosθ = xdθ……………………..(ii)
Considering triangle AOP:cosθ = r/x
∴x = r/cosθ…………………(iii)
Using the values of dlcosθ from eq.(ii) and x from eq.(iii) in eq.(i):
B = ∫(μo/4π)ixdθ/x2 = ∫(μo/4π)idθ/x = ∫(μo/4π)i(cosθ)dx/r
∴ B = (μo/4π)(sinθ2 + sinθ1)
For infinitely long wire (θ1 = 90° θ2 = 90°):The above equation becomes
∴ B = μoi/(2πr)
There is an alternative and appealing way in which the Biot-Savart law may be expressed. Ampere’s circuital law considers an open surface with a boundary. The surface has current passing through it.
The sum then tends to an integral. Ampere’s law states that this integral is equal to μ0 times the total current passing through the surface, i.e.,
where I is the total current through the surface. The integral is taken over the closed loop coinciding with the boundary C of the surface.
The solenoid and the toroid are two pieces of equipment which generate magnetic fields. The synchrotron uses a combination of both to generate the high magnetic fields required. In both, solenoid and toroid, we come across a situation of high symmetry where Ampere’s law can be conveniently applied.
Solenoid is the generic term for a coil of wire used as an electromagnet. It also refers to any device that converts electrical energy to mechanical energy using a solenoid. The device creates a magnetic field from electric current and uses the magnetic field to create linear motion.
Toroid is a hollow circular ring on which a large number of turns of a wire are closely wound. It can be viewed as a solenoid which has been bent into a circular shape to close on itself.
The force between two long straight and parallel conductors separated by a distance r can be found by applying what we have developed in preceding sections. Figure 1 shows the wires, their currents, the fields they create, and the subsequent forces they exert on one another. Let us consider the field produced by wire 1 and the force it exerts on wire 2 (call the force F2). The field due to I1 at a distance r is given to be
Moving coil galvanometer is an electromagnetic device that can measure small values of current. It is also known as Weston galvanometer.
It works on the principle that when a current loop is placed in an external magnetic field, it experiences torque, and the value of torque can be changed by changing the current in the loop
Moving coil galvanometer consists of permanent horse-shoe magnets, coil, soft iron core, pivoted spring, non-metallic frame, scale and pointer
We know that a current loop having N number of turns,and the cross sectional area A, carrying current i, when placed in and along the direction of external magnetic field B, experiences a torque given by: ԏ = NiAB
A moving coil galvanometer can be converted into a ammeter by introducing a shunt resistance rs, of small value in parallel. It can be converted into a voltmeter by introducing a resistance of a large value in series.
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