4. Moving Charges and Magnetism

Magnetic Field

Region around a current carrying conductor in which electro-magnetism effect is produced, is said to be Magnetic Field.

It is generally denoted by B. Magnetic field is a vector quantity. The dimension of magnetic field is [MA⁻¹T⁻²]. The SI unit of magnetic field is NA⁻¹m⁻¹ or Tesla.

B = F_m/qvsinθ

Here,
Fm = Magnetic Force
q = Magnitude of the charge
v = Velocity of the charge
θ = Angle between velocity and magnetic force.

Magnetic field vertically upward and downward in a plane are generally denoted by conventional sign (.) and (x) respectively. 

Total Magnetic field from different sources is the vector sum of the different magnetic field.
Therefore,

B = B1 + B2 + B3 + ...................... 
Super position principle is used during the sum of magnetic field.

Lorentz Force

The field in which both electric and magnetic field are in existence, the total force experienced by charge q due to motion is said to be Lorentz Force.

Lorentz Force = Force on charge due to electric field + Force on charge due to Magnetic field
F = Fe + Fm
Here,
F = Lorentz Force
Fe = Electric Force
Fm = Magnetic Force

Therefore,
Lorentz Force = F = qE + qvBsinθ

Special Cases of Magnetic Force

Magnetic Force = Fm = qvBsinθ
Case I
When θ = 0⁰ or 180⁰
Fm = 0
Charge will be moved either parallel or antiparallel of magnetic field.

Case II

When  θ = 90⁰ 
Fm = qvB
Charge will be moved along the perpendicular direction of magnetic field.

Magnetic Force on a current carryin conductor

                 Consider about a conductor which is placed in a magnetic field B along z axis. The direction of the magnetic field is along x axis. So that magnetic force will be along y axis according to Fleming's left hand rule.
                   We know that a large numbers of electrons are present in free state in a conductor which move opposite of current with drift velocity.

Mathematical Calculation

Let the length of the conductor = l
cross - sectional area of the conductor = A
drift velocity = v
current = I
charge = -e
No. of electrons per unit volume of the conductor = n

Now, according to Lorentz Magnetic Force
Fm = -evBsinθ
Now consider a small length dl.

Volume of conductor for this length = Adl
No. of electrons = nAdl
Charge = -enAdl

Magnetic Force for this length 
dFm = -enAdlvBsinθ

drift velocity for dl = v = -dl/dt             (The direction of dl is opposite to drift velocity)

Therefore,
dFm = -enAdlBsinθ.-dl/dt ---------------- (i)

enAdl/dt = I

Now, from (i)

dFm = I(dlxB)

For whole conductor

Fm = IlBsinθ

Direction of Fm, I and B are determined by Fleming's left hand rule.

Special Cases

Case I

When θ = 0⁰ or 180⁰
Charged particles move either parallel or anti parallel of magnetic field.

Case II

When θ = 90⁰ 

Charged particles move along the perpendicular of magnetic field.

Under this circumstances the charged particles move in circular path.

                  Consider about a charged particles of mass m and charge q moves in a circular path of radius r.

Centripetal Force experienced by the particle = mv²/r

Magnetic Force = qvB

qvBsinθ = mv²/r

r = mv/qB

Let the charged particle takes T time to move one rotation.
Therefore,
T = 2πr/v
Now putting the value of r
T = 2πm/qB

Frequency ν = 1/T = qB/2πm

Case III

When θ is other than 0⁰, 90⁰ or 180⁰

Let the angle be θ 

Magnetic field B is along x axis, current is along z axis and magnetic force Fm is along y axis.

Under this circumstances the charged particle moves in hellical path.

Vertical Components of drift velocity of charged particle = vsinθ
Horizontal Components of drift velocity of charged particle = vcosθ

Particle along vertical components moves along circular path.

Centripetal Force = m(vsinθ)²/r
Magnetic Force = qBvsinθ

Centripetal Force = Magnetic Force
r = mvsinθ/qB

T = 2πr/ vsinθ

Now putting the value of r

T = 2πm/qB

Frequency ν = 1/T = qB/2πm

Motion in Combined Electric and Magnetic Field

Consider about a charge q moves in a field in which both electric and magnetic field are perpendicular to each other.

Electric force acts along electric field and magnetic force opposite to electric force. Direction of magnetic field is determined by Fleming's right hand rule.

Mathematical Calculation

Fe = Fb

qE = qvB.sin90⁰
qE = qvB
v = qE/qB
v = E/B

Magnetic Field due to a current element, Biot Savart Law

Biot Savart Law states that 
"The magnitude of the magnetic field is proportional to the current, the element length and inversely proportional to the square the distance of the point from current element at which magnetic field is determined. The direction of the magnetic field is perpendicular to the plane containing element length and distance."

          Consider about a finite conductor XY carrying current I. dl is the current element. There is a point P at a distance r from the current element. At this point magnetic field dB is to be determined. The angle between dl and displacement vector r is θ.

The relevant figure is drawn below

Now according to Biot Savart Law

dB∝ Idlxr/r³
dB = 𝛍ₒIdlxr/4𝛑r³

Here 
𝛍ₒ/4𝛑 is a constant of proportionality ( The medium is vacuum)

Therefore, magnitude of the magnetic field
।dB। = 𝛍ₒIdlsinθ/4𝛑r²

The value of 𝛍ₒ/4𝛑 = 10⁻⁷ Tm/A
𝛍ₒ is the permeability of free space or vacuum.

There are some similarities, as well as differences, between Biot Savart's Law and Coulomb's Law. Which are as follows

1. Both depend inversely on the square of distance from the source of interest.
2.The principle of superposition applies to both magnetic and electric fields.
3. Both magnetic field and electric field is linear to their sources Idl (current element) and q (source of electric charge) respectively.
4. The electric field is produced by scalar source q (electric charge) whereas magnetic field is produced by vector source Idl (current element)
5. The electric field is along the displacement vector whereas magnetic field is perpendicular to the plane containing the displacement vector r and current element Idl.
6. The magnetic field at any point in the direction of dl is zero.

Relation between permitivity of free space (𝛆ₒ) and permeability of free space (𝛍ₒ) and speed of light (c).

𝛆ₒ𝛍ₒ = 4𝛑𝛆ₒ.𝛍ₒ/4𝛑

= 10⁻⁷/9x10⁹
= 1/9x10¹⁶
= 1/(3x10⁸)²
= 1/c²

Chapter 4 Class XII