Introduction:

Like In Electrostatics we have coulomb's law to find electric fields due to charge distribution, but that is very tedious to use in many cases. So in case of some symmetry, we had Gauss's law that can be used to find electric fields in an easier way.

In the same manner in Magnetostatics we have the Biot-Savart law that can be used to find magnetic fields due to any current distribution.

But in the case of some symmetry we have, we can use ampere circuital law that can make our life easier.

Ampere circuital law

Ampere circuital law states that the line integral of the magnetic field  B around any closed circuit is equal to μ0 ( permeability constant) times the total current ‘I', Threading or passing through this closed circuit.

Mathematically,

 B dl = μo* I

Where I is the net current enclosed by the closed circuit. The closed curve is called the Amperean loop, which is a geometrical entity and not a real wire loop.

Ampere circuital law can be used to find magnetic field due to magnetic field in some symmetry case like long wire, circular loop, cylindrical conductor etc,

Above are some examples of magnetic fields in some standard cases. We will discuss Solenoid and toroid in detail

Force on a current carrying conductor in a magnetic field

When the conductor carrying current is placed in an external magnetic field, it experiences a mechanical force. The direction of the force is perpendicular to both the current and the magnetic field and it is given by Fleming’s right left-hand rule.

Cause of the force: A current is an assembly of moving charges and magnetic field exerts a force on moving charges, That is why a current-carrying conductor experiences a side-ways force as the force experienced by the moving charges( free electrons) is transmitted to the conductor as a whole.

The expression of the force will be derived as

Thus Force on the current carrying conductor placed in a magnetic field is given by F= IlBsinθ= I ( l×B)

Direction of force is given by Fleming’s left-hand rule.

If we place the fingers of our left hand in mutually perpendicular directions. If the forefinger is along the direction of the magnetic field and the middle finger points in the direction of current then the thumb will give the direction of the force on the conductor.

The force between two parallel current

As we know, the current carrying conductor produces a magnetic field around it and when any current carrying conductor is placed in a magnetic field it will experience a force on it.

So if we have one current carrying conductor it will produce a magnetic field around it. If we place another current carrying conductor near the first conductor, it will actually be a current carrying conductor placed in the magnetic field and hence will experience a force on it.

In the figure given above It is shown that two parallel current carrying conductors placed near each other exert force on each other.

  • When they have parallel currents, they attract each other.
  • When they have antiparallel current, they repel each other.
  • The magnitudes of forces F1 and F2 are equal but they are in opposite directions which is in accordance with Newton’s third law.

The force between two parallel current carrying conductors.

Consider two conductors having parallel currents  I1, I2  are placed at a distance ‘r’ from each other. Magnetic field of one conductor at the place of the second conductor is B1=μo* I1/(2Πr)

Force on 2nd conductor having length ‘L’  due to 1st conductor= F21F21=I2*L*(B1)

Force on 1st conductor due to 2nd conductor = F12

F12= - F21

Force per unit length  on each conductor F= μo*I1*I2/ (2Πr)

Similarly, force on two parallel conductors having antiparallel currents is shown below. Conductors having ant parallel currents repel each other.

Force on two straight wires having ant parallel currents.

Torque on a current loop in a uniform magnetic field.

The figure shows a rectangular loop carrying a steady current I and placed in the uniform magnetic field B. In this discussion, we will see that the current loop placed in uniform magnetic field experiences no net force but experiences a torque.

Initially when the rectangular loop is placed such that Uniform magnetic field B is in the plane of the loop. Then Side AD and BC would be parallel to magnetic field B

F= I ( l×B)= IlBsinθ

Therefore force on side AD and BC is zero.

F4=F3=IlBsin0=0

Force on AB and CD is maximum as these sides are perpendicular to magnetic field B and also equal in magnitude.

F1=F2= IlBsin90= IlB 

F1 is directed into the plane of paper and F2 is directed out of the plane of the paper. Thus the net force on the loop is zero.

Thus the net force on the loop is zero.

There is a torque on the loop due to a pair of forces F1 and F2. The torque on the loop tends to rotate it.

Torque= Force * perpendicular distance

τ=F1*(d/2)+ F2*(d/2)

τ= (IlB)*(d/2) + (IlB)*(d/2)=I(ld)B= IAB

Where A= ld is the area of a rectangular loop.

When the plane of the loop makes an angle ‘θ’ with the uniform magnetic field B.

Then Force on Side BC and DA will be equal and opposite as their currents are in opposite directions. There is no net force and torque due to force on side BC and DA as these forces are collinear along the axis and hence cancel each other.

Force on arm AB and CD are F1 and F2, They too are equal and opposite in magnitude but they are not collinear and hence constitute a torque.

F1=F2=IlB

The torque on the loop, in this case, is however less than the torque when the loop was placed in the plane of the magnetic field B.

τ= F1(a/2)sinθ+ F2(a/2)sinθ=Il aBsinθ= IABsinθ

When angle ‘θ’ tends to zero, the perpendicular distance between them also approaches zero, Thus making the force collinear and the net force and net torque zero.

For N turns τ=NIABsinθ 

Also, τ=mBsinθ= m×B

Where m= NIA magnetic moment of N turns coil.

Magnetic moment is a measure of an object's tendency to align with a magnetic field. It is a vector quantity.

m= current* Area= I A

The Magnetic dipole Moment of revolving electron

In Bohr’s Model of atom, electrons revolve in circular orbits around the positively charged nucleus under electrostatic force just the way planets revolve around the sun under gravitational force.

Thus the electrons of charge (-e) perform uniform circular motion around a stationary heavy nucleus (+Ze), This constitutes a current ‘I’.

The time period of the electron around circular orbit with uniform speed ‘v’ is given by

The circulating current ‘I’ is given by

Moving coil galvanometer

Moving coil galvanometer is an instrument used for deflection and measurement of small electric currents and voltages.

Principle: It is working is based on the fact that when a current-carrying coil is placed in a magnetic field. It experiences a torque.

  • Moving coil galvanometer consists of a coil with many turns, free to rotate about a fixed axis in a uniform radial magnetic field.
  • There is a cylinder with a soft iron core which not only makes the field radial but also increases the strength of the magnetic field.

  • When current flows through the coil, torque NIBA acts on it. The magnetic torque tends to rotate the coil by angle Φ. The spring provides counter-torque kΦ that balances the magnetic torque NIAB, resulting in angular deflection Φ.
  • In equilibrium   KΦ= NIBA, where K= torsional constant of spring ( restoring torque per unit twist)
  • Deflection Φ= (NAB/k) *I, Thus deflection is directly proportional to the current in the coil so it is used to detect and measure currents.

  • Deflection per unit current (Φ/ I)= NAB/k is called the sensitivity of the galvanometer.
  • A convenient way to increase the sensitivity of the galvanometer is to increase the number of turns.

Galvanometer as an ammeter and Voltmeter

A Galvanometer can be used as an ammeter to measure the value of the current.

  • For measuring currents, the galvanometer has to be connected in series but the resistance of the galvanometer is very high, this will change the current in the circuit.

  • To overcome this situation a shunt resistance ‘S’ is attached parallel to the galvanometer coil to drastically reduce the resistance of the galvanometer.
  • The scale of this ammeter is calibrated and graduated to read off the current values.

The galvanometer can be used as a voltmeter to measure the voltage across a given section of the circuit.

  •  For this galvanometer must be connected parallel with the section of the circuit whose potential difference is to be measured.
  • It must draw a very small current from the circuit, otherwise it will disturb the original voltage.

  • To ensure that the voltmeter draws negligible current from the circuit, the resistance of the voltmeter must be very high. Therefore a very high resistance is connected in series with a galvanometer to make it work like a voltmeter.

 

 

 

 

 

 

 

 

 

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