2. Introduction:
Oersted Experiment:
Magnetic field and Intensity:
The magnetic field is the region around a magnetic material or a moving electric charge
within which the force of magnetism acts.
The strength of the magnetic field is called Magnetic intensity.
A current is flow through a conductor in circuit.
If a magnetic compass is brought near the circuit
then it get deflected.
So there is a magnetic field produced by the
conductor.
Moving charge always create a magnetic field.
In magnet similar pole repel each other and opposite pole attract each other.
The force between two magnet are called magnetic force.
There are several experiments and phenomenon were observed by scientise.
A hanging magnetic needle always pointed towards north pole of earth.
Earth has it’s own magnetic field.
3. Biot-Savart Law:
Biot–Savart law is an equation describing the magnetic field generated by a constant electric current.
It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.
It used for computing the resultant magnetic field B at position r in 3D-space generated by a current I caring
conductor.
According to Biot-Savart law, magnitude of magnetic field dB at point P due to current element depends on-
𝑑𝐵 ∝ 𝐼 𝑑𝐵 ∝ 𝑑𝑙 𝑑𝐵 ∝
1
𝑟 − 𝑟/ 2
𝑑𝐵 ∝ sin 𝜃
𝑑𝐵 =
𝜇0
4𝜋
𝐼𝑑𝑙 sin 𝜃
𝑟 − 𝑟/ 2
𝒅𝑩 =
𝝁𝟎
𝟒𝝅
𝑰𝒅𝒍 × 𝒓 − 𝒓/
𝒓 − 𝒓/ 𝟑
The direction is ⊥to the plane containing 𝐼𝑑𝑙 & 𝑟 − 𝑟/. Here by Right Hand rule direction of field is inward
to the screen.
𝐵 = 𝑑𝐵 =
𝜇0
4𝜋
𝐼𝑑𝑙 × 𝑟 − 𝑟/
𝑟 − 𝑟/ 3
Due to whole conductor field at P
4. Magnetic field due to straight conductor:
This law is use to determine magnetic field at any point around a current caring conductor.
The magnetic field for long straight conductor at P will be-
∵ As shown in fig; 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 180 − 𝜃 =
𝑟
𝑟2 + 𝑧2 and 𝑟 − 𝑟/ 2
= 𝑟2 + 𝑧2
∴ 𝑑𝐵 =
𝜇0
4𝜋
𝐼𝑑𝑙 sin 𝜃
𝑟 − 𝑟/ 2
𝐵 =
−∞
∞
𝜇0
4𝜋
𝐼𝑑𝑙𝑟
𝑟2 + 𝑧2
3
2
𝐵 =
𝜇0𝐼
2𝜋𝑟
5. For a circular current loop:
𝑑𝑙 × 𝑟 = 𝑑𝑙𝑟 sin 90 = 𝑑𝑙𝑟 & 𝑟 cos 𝜃 = 𝑅 & 𝑟 = 𝑅2 + 𝑧2
Resolving the 𝑑𝐵 and 𝑑𝐵/ into rectangular component at
point P, it is clear that vertical component will be equal and
opposite, hence cancel each other. But horizontal component
will be added up and act in the direction OY.
Hence using Biot-Savart law
𝑑𝐵 =
𝜇0
4𝜋
𝐼𝑑𝑙 × 𝑟
𝑟3
cos 𝜃
𝑑𝐵 =
𝜇0
4𝜋
𝐼𝑑𝑙𝑅
(𝑅2 + 𝑧2)
3
2
Hence magnetic field for whole circular conductor will be
𝐵 =
𝜇0
2
𝐼𝑅2
(𝑅2 + 𝑧2)
3
2
; in the direction of OY
6. Lorentz law:
A moving charge in a magnetic field always experience a force on it.
The force in the direction perpendicular to the plane of magnetic field and velocity of charge.
The maximum force act on the charge when it move perpendicular to magnetic field.
No force act on a charge which moves parallel to the magnetic field.
Also no force act on a static charge.
If charge move in perpendicular direction to field then it trace a
circular path.
But if it move at any angle 0<X<90 then it trace a helix.
If a charge move in a combination of electric and magnetic
fields then the net amount of force will be-
𝐹𝑚𝑎𝑔 = 𝑞(𝑉 × 𝐵)
𝐹𝑇𝑂𝑇 = 𝐹𝐸𝐿𝐸 + 𝐹𝑚𝑎𝑔
𝐹𝑇𝑂𝑇 = 𝑞(𝐸 + 𝑉 × 𝐵)
If many charges are present then- 𝐹𝑚𝑎𝑔 = 𝑛𝑞(𝑉 × 𝐵)
𝐹𝑚𝑎𝑔 = 𝑗 × 𝐵
7. Force between current elements:
No force act on a current caring conductor if they are
perpendicular to each other and also if they are in the same
line in the same direction.
If we have two current element the force on each other will be-
𝐹12 =
𝜇0
4𝜋
𝐼1𝑑𝑙1 × 𝐼2𝑑𝑙2 × 𝑟12
𝑟12
3
𝐹12 = 𝐼1𝑑𝑙1 × 𝐵
For many current element, total magnetic field will be
𝐵 =
𝜇0
4𝜋
𝑗 × 𝑟12
𝑟12
3 𝑑𝑉
9. Magnetic Flux:
How much of something is flowing through a given area at per unit time called flux.
If we have a surface place at a magnetic field at an angle then the amount of magnetic field line passing
through the surface per unit time per unit area is called magnetic flux.
Φ = 𝐵𝐴 cos 𝜃
Φ = 𝐵. 𝐴
𝑑𝜙 = 𝐵. 𝑑𝐴
Φ = 𝑑𝜙 = 𝐵. 𝑑𝐴
If magnetic field is not uniform over the surface
10. Divergence of magnetic field:
Field lines of a bar magnet always start from north pole and end at south pole.
Similarly the magnetic field of current caring conductor always around it.
No matter how small a magnet is but it always have north and south pole.
If we take a close surface around a conductor or magnet, then amount of
flux enter the surface is always equal to leaving the surface.
∇. 𝐵 = 0
So B always circulates don’t diverge.
There is no point source of magnetic field lines.
So magnetic monopole don’t exist.
This is also called Gauss law of magnetism.
𝐵. 𝑑𝐴 = 0
11. Curl of magnetic field:
Magnetic field lines starts from north pole and end at south pole.
For current caring conductor field is around the conductor.
So magnetic field has a curl.
Curl can calculate using Ampere law.
It states that, for any closed loop path, the sum of the length elements times the magnetic field in the
direction of the length element is equal to the permeability times the electric current enclosed in the loop.
𝑩. 𝒅𝒍 = 𝝁𝟎𝑰𝒆𝒏𝒄
𝐼𝑒𝑛𝑐 = Total current enclosed by integration path.
If flow of charge is represented by a volume current density J-
Then; 𝐼𝑒𝑛𝑐 = 𝐽. 𝑑𝑎
The integral taken over the surface bounded by the loop
Applying Stokes theorem-
𝐵. 𝑑𝑙 = ∇ × 𝐵 . 𝑑𝑎 = 𝜇0 𝐽. 𝑑𝑎
𝜵 × 𝑩 = 𝝁𝟎𝑱
12. Boundary Condition
Magnetic field is discontinuous at boundary of two medium as
the electric field.
We have a surface current element and magnetic field is
passing at angle.
Magnetic field have two component- perpendicular & parallel.
We have to check discontinuity for two component separately.
For Normal component:
As per Gauss law of magnetism- 𝐵. 𝑑𝐴 = 0
𝐵1Δ𝑋Δ𝑌 − 𝐵2Δ𝑋Δ𝑌 = 0
𝑩𝟏 = 𝑩𝟐
𝑇𝑜𝑝
𝐵. 𝑑𝐴 +
𝐵𝑜𝑡𝑡𝑜𝑚
𝐵. 𝑑𝐴 = 0
So Normal component is continuous over the boundary.
13. For Tangential Component:
According to Ampere’s Law if an Amperian loop
running perpendicular to a current then-
𝐵. 𝑑𝑙 = 𝜇0𝐼𝑒𝑛𝑐
𝐴
𝐵
𝐵. 𝑑𝑙 +
𝐵
𝐶
𝐵. 𝑑𝑙 +
𝐶
𝐷
𝐵. 𝑑𝑙 +
𝐷
𝐴
𝐵. 𝑑𝑙 = 𝜇0𝐼𝑒𝑛𝑐
𝐵𝑡1Δ𝑋 − 𝐵𝑡2Δ𝑋 = 𝜇0𝐾Δ𝑋
Where K is the current per unit length
𝑩𝒂𝒃𝒐𝒗𝒆 − 𝑩𝒃𝒆𝒍𝒐𝒘 = 𝝁𝟎 𝒌 × 𝒏
Where 𝑛 is the unit vector perpendicular to the surface pointing upward.
𝑩𝒕𝟏 − 𝑩𝒕𝟐 = 𝝁𝟎𝑲
So by combining the parallel & perpendicular component we get-
So Tangential component is discontinuous over the boundary.
