What is Magnetic flux?It is defined as the number of magnetic field lines which pass through the close surface. It is the total magnetic field passing through the surface area. The area of magnetic field which will consider can be of any size and may be under any orientation with respect to the direction of magnetic field.

**Symbol **

The symbol of magnetic flux which use commonly is a Greek letter Phi ⏀or Phi suffix B ⏀_{B}

**Formula **

The formula is given below

⏀_{B}=B.A=BACosƟ

In the above equation

- ⏀
_{B}=magnetic flux - B=Magnetic field
- A=Area
- Ɵ=angle at which field line pass through the given surface area

**Unit **

To measure magnetic flux we can use fluxmeter commonly. The SI and CGS unit is Weber(Wb) and Maxwell respectively.

- SI unit is Weber(Wb)
- CGS unit is Maxwell.

Fundamental unit is Volt-seconds

**Magnetism**

A magnetism is a phenomenon which is related to the magnetic field, which arise from the motion of electric charges and due to this motion field produced which attract or repel other objects.

A magnet (from Greek magnetization stone) is a material that produce a magnetic field .This magnetic field is invisible but is responsible for the most notable property of a magnet: a force which pull other ferromagnetic material such as iron,, and attract or repel other magnets also. Permanent magnet is a material which is magnetize and create its own persistent magnetic field. An everyday example is a refrigerator magnet used to hold notes on a refrigerator door

**Brief concept about the magnetic flux**

The Great scientist Faraday conduct several experiment on electromagnetic induction. Due to their wide contribution in science he is known as greatest experimental scientist of nineteenth century. Magnetic flux play vital role in electromagnetic induction therefore it is important to understand the magnetic flux .

**When magnetic field is Uniform**

In order to calculate the magnetic flux ,firstly we consider the magnet which have magnetic field B and the area A .The ,magnetic flux through the given plane area A in a uniform magnetic field is given as the scalar product of the magnetic field B and the area A.In this equation the angle at which the field line passing through the surface is also very important,There are two possible angle which is very important

- When the angle between the magnetic field vector and the area vector is nearly equal to 90
^{0}then the resulting flux s very low - When the angle between the magnetic field and area is equal to 0
^{0},then the resulting flux will be maximum.

Mathematically the equation is given by

**⏀ _{B}=B.A=BACosƟ**

In the above equation Ɵ is the angle between vector A and B.

**When the magnetic field is non-uniform.**

When the magnetic field is non-uniform which mean that at different parts of the surface the magnetic field is different in magnitude and direction.then the total magnetic fluxpassing through the surface area is the summation of the product of all the area and their corresponding magnetic field

Mathematically it is given by

**⏀ _{B}=B_{1 } .dA_{1}+ B_{2 } .dA_{2}+ B_{3 } .dA_{3}+……………………..∑_{all} B_{i } .dA_{i}**

In the above equation it is clear that the mantic flux is a scalar quanitity.The SI unit of mantic flux is Weber( Wb) or Tesla meter square (T.m^{2})

**How we measure magnetic flux**

The magnetic flux can be measured with the help of magnetometer. And is unit us Weber or tesla meter square which is named after German physicist Wilhelm Weber.

**Magnetic flux Density**

The magnetic flux density is define as the amount of magnetic flux in the given area which is taken perpendicular to the magnetic flux’s direction.

Or

Magnetic flux density is also defined as the force acting per unit length on a wire which is place at right angles to the magnetic field.

Or

The magnet flux per unit area is called magnet flux density .

B =⏀/A

So ,

1T = 1Wb m^{-2}

Magnetic flux density is a vector quantity and it is denoted by B.The unit of magnetic flux density is Tesla or gausses.The magnetic flux density is also called magnetic induction.

Unit of magnetic flux density | |

SI System unit | Tesla Abbreviated as T |

CGS system unit | Gauss which is abbreviated as G or Gs |

**Force on a current carrying conductor**

When a current carrying wire is placed at the right angle to a uniform magnet field ,the magnetic field of the wire and the external uniform field interact, resulting in the force F on the wire . This force depends on the current I on the wire , and the length L of the wire that lies in the field .

F & IL

The strength of the uniform field , which is called magnet induction B, is the constant of proportionality .

F=BIL

If the wire is place in the field B at some angle Ɵ with it , then the force is given by the relation

F=BILsinƟ

Or n vector form

F=(L X B)

This equation shows the magnitude of force F on a current Carrying wire depend upon the following factors ;

B ;Magnitude of magnetic induction

I; amount of current flowing in the wire

L; length of the wire lying in the field

Ɵ; angle between B and L

This force F is maximum when Ɵ is 90 and minimum, when the Ɵ is 0 , if we keep B ,L and L constant .

As force is vector quantity so its direction should also determined the direction of F s determined by Fleming^{,}s left hand rule

The thumb and the first two fingers of left hand are set at right angle to each other.

With the first finger pointing in the direction of the field and the second finger pointing in the direction of current , the thumb gives the direction of the force . The S.I. unit for induction B is tesla T and it can be defined as

B=F/IL

1T=1N/Am

If the force experienced by 1m of wire carrying 1A current placed perpendicularly in magnetic field is one Newton , then the magnetic induction is one Tesla .

The another unit of B is Gauss given by

1G=10^{-4 }T

Or 1T = 10^{4} G

**MEGNET FLUX DUE TO A CURRENT CARRYING SOLENOID**

A solenoid is long spring like coil of length many times its diameter with many turn s every centimeter. A current I in the solenoid produce a magnetic field B along its axis . the magnetic of the solenoid is strong along its axies and weaker , rather negligible outside.

To determine the value of B of solenoid let us consider an amperian path abcd with length l_{1}and l_{2} much longer as compare to other two lengths . Now applying ampere law ,

∑B.⧍L =µ_{0} ẖ

B.l_{1}+B.l_{2}+B.l_{3}+B.l_{4}= µ_{0 }l

Now inside the solenoid ,B and l_{1} are parallel,so B.l_{1}=Bl_{1}

Outside B=0 ,so B.l_{3}=0

For l_{2} and l_{4} B and lengths are perpendicular ,so B.l_{2}=0 and B.l_{4}=0

Therefore the field for a solenoid is given by

∑B.⧍L =B.l_{1}= µ_{0 }l

Now if L= length of the solenoid and

N=total number of turns in the solenoid then

∑B.⧍L =BL=N µ_{0 }l

So B=n µ_{0 }l,

Where n=N/L Number of turns per unit length.

Example :A solenoid is 10 cm long and is wound with two layers of wire .the inner layer has 50 turn and the outer layer has 40 turns.A current of 3A flows in both layers in the same direction.What is the magnitude of magnetic flux density along the axis of solenoid .

Solution: Length of a solenoid =L=10cm

Inner layer=n_{1}=50cm

Outer layer=n_{2}=40 cm

Flow of current=I=3A

Magnetic flux density=B=?

B=n_{1}µ_{0}l_{1}+ n_{2}µ_{0}l_{2}

=3.4×10^{-3} T

**Ampere’s Law**

Ampere’s Circuital law which is discovered by Andre Marie Ampere in 1826 ,is related to the integrated magnetic field in a loop around a current carrying wire to the current passing through the wire. We know that when a current flow through the wire ,a magnetic field establish around the wire .If we consider a close path around the wire in the form of a circle having the wire at the center, then the magnitude of magnetic flux density B changes with the current I in the wire and the distance r from the wire .

So

B ⍺ I

And B ⍺ 1/r

Or B ⍺ I/r

Summing up all around the circular path

B=µ_{0} I/2π r

µ_{0}=Permeability of free space (4π x10^{-7} Wb A^{-1} m^{-1})

Now consider a closed path around the wire .For any path element ⧍L we can write

B. ⧍L= µ_{0}I

As B and ⧍L are parallel ,so B. ⧍L=B⧍L= µ_{0}I

Now summing over the entire close path ,

∑B . ⧍L= µ_{0}I

Which is Ampere’s Law. The closed path is called the amperian path. In general this law can be applied to any path around a uniform magnetic field.

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