Magnetic Field

Magnetic Field: The space or region around a magnet (moving charge or current-carrying conductor) up to which its magnetic influence can be experienced is called the magnetic field.
The strength of the magnetic field at any point in a field is called the magnetic field intensity. It is denoted by vector B and its SI unit is Tesla (T).

Magnetic Field Lines

If an isolated N-pole of a bar magnet is moved in the direction of repelling force acting on it, the isolated pole will trace out a line called magnetic field line.

• The tangent drawn to magnetic field line gives the direction of the magnetic field.
• The closeness or density of field lines is directly proportional to the strength of the field.
• Magnetic field lines appear to emerge from the N-pole and terminate at the S-pole.
• Inside the magnet, the direction of magnetic lines is from S-pole to N-pole.
• Magnetic field lines never intersect with each other.
• Magnetic field lines form a closed loop.
• Field lines have both direction and magnitude at any point in the field, so magnetic field lines are represented by a vector.
• The magnetic field is stronger at the poles because the field lines are denser near the poles.

Force on Moving Charge Inside Magnetic Field (Lorentz Force)

Consider a charged particle having charge $q$, moving with velocity v in a magnetic field B. Let the magnetic field be directed along the x-axis, and the particle moves making an angle $\theta$ with the x-axis. Then the charged particle experiences a magnetic force F, which is directly proportional to:

Force on Current Carrying Conductor:

Consider a conductor of length $l$ and cross-sectional area A placed at an angle in a uniform magnetic field B. Let I be the current passing through the conductor, vd be the drift velocity of free electrons, n be the electron density, and e be the charge of an electron. Then, the force on each electron is:

Special cases:

1.When θ=0∘or 180∘, F=0

∴ No force is experienced by conductor when placed parallel or anti-parallel to magnetic field.

2.When θ=90∘, $=BIl$, i.e. maximum force

Force and Torque on a Current Carrying Rectangular Coil in Uniform Magnetic Field

Consider a rectangular coil PQRS carrying current $I$, is suspended in horizontal uniform magnetic field $B$ so it rotates freely about vertical axis XY as shown in the figure. Let, $PQ=RS=l$, and OR=PS=b

Let at any instant, the plane of coil makes an angle $\theta$ with the direction of magnetic field. The force on various arms of the coil is given by:

      (4) The upward force acting on arm SP:

F₂ = BILbsinθ

It is seen that forces F₂ and F₄ are equal in magnitude but opp. in direction and pass through same line of action. so, they cancel each other.

Forces F₁ and F₃ are also equal in magnitude and opposite in direction but do not passed through same line of action and cause to rotate the coil about a vertical axis.

Moment of couple or torque, τ is given by
τ = magnitude of either force × perpendicular distance between forces

= F₁ × bsinθ
= BIL × bsinθ
= BIA sinθ

where A = l × b is the area of the coil. Therefore, torque
     τ = BIA sinθ

If the coil has N turns, then
τN = BIAN sinθ

If the plane of the coil makes an angle θ’ with the magnetic field, then the angle made by the side PS or QR with B is θ + θ’ = 90° or θ = 90 – θ’. Then,
τN = BIAN sin (90 – θ’)
∴ τN = BIAN cosθ’

Moving Coil Galvanometer

Moving coil Galvanometer is an instrument which is used to measure the electric current. It is a sensitive electromagnetic device which can measure low current even of the few microamperes.

It is mainly divided into two types:

• Suspended coil galvanometer.
•  Pivoted coil or Weston galvanometer.

Principle: When a current carrying loop or coil is placed in a uniform magnetic field, the coil experiences a torque.

Construction:
The instrument contains a rectangular coil having large no. of turns wound on a non metallic frame. The coil is suspended between two poles of a permanent magnet which are cylindrical in shape. The coil is suspended by a phosphor bronze strip which acts as a path for the current to the coil which is finally connected to terminal T₁ of the galvanometer. The other end of the coil is connected to a light spring which is finally connected to the terminal T₂ as shown in figure. The spring exerts a very small

Sensitivity of Galvanometer

A galvanometer is said to be sensitive if a small amount of current flowing through its coil produces a large deflection in it.

Current Sensitivity: The current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer per unit current flowing through it.

Where:

• $\phi$ = Deflection produced
• $B$ = Magnetic field strength
• $N$ = Number of turns in the coil
• $L$ = Length of the coil
• $I$ = Current flowing through the coil
• $k$ = Constant related to the galvanometer’s construction

Current sensitivity can be increased by:
i) Increasing B
ii) Increasing N
iii) Increasing L
iv) Decreasing k

Voltage Sensitivity: Voltage sensitivity of a galvanometer is defined as the deflection produced in the galvanometer per unit voltage applied to it.

Where:

• $\phi$ = Deflection produced
• $B$ = Magnetic field strength
• $N$ = Number of turns in the coil
• $L$ = Length of the coil
• $V$ = Voltage applied
• $k$ = Torsional constant
• $R$ = Resistance of the coil

Voltage sensitivity can be increased by:
i) Increasing B
ii) Increasing N
iii) Increasing L
iv) Decreasing k
v) Decreasing R

Hall Effect: 

When a magnetic field is applied to a current carrying conductor, a voltage is set up in the direction perpendicular to both the direction of current and magnetic field. This effect is called Hall Effect.

Let us consider a slab of width $w$ and thickness $t$ so that the area of cross section is t×d . Suppose conventional current flows towards right so that free electrons move to left with drift velocity v_d as shown.

When a uniform magnetic field is applied from face TUVW to front face PQRS, each electron experiences a Lorentz force, F=qv_d in vertically downward direction. So, electrons accumulate to lower face VWSR which becomes negatively charged and upper plate positively charged. So, pd or emf is developed across lower and upper face of slab which passes electron flow towards downward direction. The flow ceases when emf reaches its critical value.

Biot–Savart Law

The Biot-Savart Law is a fundamental equation in electromagnetism that describes the magnetic field generated by a current-carrying conductor

Consider a conductor XY through which a current I is flowing. The current produces a magnetic field around it. To find the magnetic field at a point P, consider a small element AB of length $d\ell$ which makes an angle $\theta$ with the line joining it to point P. Let the distance from the element to point P be r.

According to the Biot–Savart law, the magnetic field $dB$ produced by this element at point P is:

• directly proportional to the current flowing through the conductor,       

dB  ∝I

• directly proportional to the length of the current element,

dB ∝  dl

• directly proportional to the sine of the angle between the current element and the line joining the element to the observation point,

dB ∝ sinθ

• inversely proportional to the square of the distance between the element and the observation point.

.Combining the Relations, mathematically, it is expressed as:

where k is the proportionality constant. Value of the Constant is 

where μ0 is the permeability of free space, and its value is μ₀=4π×10⁻⁷ H/m.

Direction: The direction of the magnetic field $\mathbf{dB}$ at point $P$ is perpendicular to the plane containing the element $d\ell$, and the position vector $\mathbf{r}$.

Applications of Biot and Savart Law:

There are several applications of Biot and Sevart, and some of them are described below. 

 Magnetic field at the centre of a current-carrying circular coil.

Consider a circular coil of radius r carrying current I, which produces a magnetic field around it. To find the magnetic field at O, consider an element XY of length dℓ such that the radius r = r makes angle θ = 90° with the element. From Biot, and Savart law, dB produced by this element at O is

 Magnetic Field on the Axis of  Current-Carrying Circular Coil.

Compiled by: Er. Basant Kumar Yadav