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.

Oersted discovery

Magnetic effect of electric current was first discovered by Oersted in 1820 AD. According to him, when a current is passed through a conducting wire placed above and parallel to the axis of a compass needle, the needle is deflected.

When the current is passed through the wire from south to north, the north pole of the compass is deflected towards the west and when the current is passed through the wire from North to south, the north pole is deflected towards East as shown in the figure.

Direction of electric current and Magnetic field

The direction of the magnetic field produced by the current carrying conductor is given by following rules:

a) Right-hand thumb rule (right-hand grip rule)

The right-hand thumb rule gives the direction of the magnetic field due to a straight current-carrying conductor. According to this rule, when the current-carrying conductor is gripped by the right hand in such a way that the thumb points the direction of the flow of current, then the curled fingers show the direction of the magnetic field.

b) Right-hand fist rule

The direction of the magnetic field due to a circular current-carrying conductor is given by the right-hand fist rule. According to this rule, the fingers of our right hand are curled in the direction of the current in the circular conductor, and then the direction of the magnetic field is given by the direction of the thumb.

Fleming’s left-hand rule

When a current-carrying conductor is placed in a magnetic field, it experiences a force called the Lorentz force. It states that, “if the thumb, fore finger, and middle finger of our left hand are stretched in mutually perpendicular directions, such that the fore finger points the direction of the magnetic field and the middle finger points the direction of the current, then the thumb points the direction of the force experienced by the conductor.

Force on a moving charge in a uniform magnetic field

Suppose a charge q is moving with uniform velocity v in a uniform magnetic field B. It is found that the charge experiences a force F along the x-axis, such that the force is directly proportional to

a) The magnitude of charge q
  F ∝ q ——– (1)
b) The speed v of the charge

F ∝ v ———–(2)
c) The applied magnetic field B
  F ∝ B ———–(3)
d) the sine of angle between the direction of v and B
  F ∝ Sin θ ———(4)

On combining, we get

  F ∝ B q v Sin θ
  F = k B q v Sin θ

Where k is the proportionality constant. Experimentally, it is found to be 1.

So, F = B q v Sin θ ——-(5)

In vector notation,

Equation (5) gives the magnitude of the force acting on the charge particle moving in a magnetic field.

Special cases:

a) when θ = 0° or 180°, then F = B q v Sin 0° = 0
 Therefore, the charge particle doesn’t experience any force when it moves
 parallel or antiparallel to the magnetic field.

b) when θ = 90°, then F = B q v Sin 90° = B q v
 Therefore, the charge particle experiences maximum force when it moves
 perpendicular to the magnetic field.

c) when v = 0, then F = 0
 That means the charge particle at rest does not experience any force.

d) when q = 0, then F = 0
 That means the neutral particle moving in a magnetic field does not experience any force.

Lorentz force

The total force experienced by a moving charged particle when both electric and magnetic fields are present is called the Lorentz force. The value of the Lorentz force

Force on a current-carrying conductor in a magnetic field

When a current-carrying conductor is placed in a magnetic field, the charges moving within the conductor experience a magnetic force, and this force is transmitted to the material of the conductor as a whole. Thus, the current carrying conductor experiences the magnetic force.

Let us consider a straight current-carrying conductor of length l and cross-sectional area A carrying a steady current I. It is placed in uniform magnetic field of flux density B which is inclined at an angle θ with the direction of current.

We assume the conventional direction of current and hence moving charges are positive. Let v_d be the drift velocity of the moving charges along the direction of current. The magnetic Lorentz force experienced by each charge of magnitude q = e is given by

F=B_ev_d​sinθ

If there are n number of free charges per unit volume of the conductor, then

total free charge = n × volume of the conductor
       = nAl

The magnetic force experienced by the conductor is the total magnetic force experienced by all the free charges

Therefore,  F = n Al × B e v_d​sinθ

                                = v_d e n A B I Sin θ

Here, v_d e n A = I, therefore, the magnetic force on the current-carrying conductor becomes

F = B I l Sin θ

In vector form,

The direction of force on the conductor due to the magnetic field is given by
Fleming’s left-hand rule.

Cases:

a) When θ = 0° or 180°, F = B I l Sin θ = 0
Thus, the current-carrying conductor does not experience any force when it is placed along the magnetic field.

b) When θ = 90°, F = B I l Sin θ = B I l (max)
Thus, the current-carrying conductor experiences maximum force when placed perpendicular to the magnetic field.

Torque a current-carrying rectangular coil

Consider a rectangular coil PQRS carrying current I, placed in the magnetic field of strength B as shown in Figure (a). Let the length of coil PQ = RS = l and the width QR

= PS = b. Coil can rotate about a vertical axis, and the magnetic field is taken as horizontal.

Let at any instant coil makes an angle θ with the magnetic field B. The force acting
on various arms of the coil are given by :

1. Force F₁ acting on side PQ is given by.
     F₁ = B I PQ sin 90°
     F₁ = B I L 1

  F₁ = B I L  ………………… (1)
  This force acts in the outward direction.

2. Force F₂ acting on side QR is given by,
     F₂ = B I QR sin θ

  F₂ = B I b sin θ  …………… (2)
  This force acts in the downward direction.

3. Force F₃ acting on side RS is given by,
     F₃ = B I RS sin 90°
     F₃ = B I l 1

  F₃ = B I L  ………………… (3)
  This force acts in an inward direction.

4. Force F₄ acting on side PS is given by,
     F₂ = B I PS sin (180° – θ)

  F₂ = B I b sin θ  …………… (4)
  This force acts in an upward direction.

The forces F₂ and F₄ are of equal magnitude but act in opposite directions on two sides of the coil. They act on the same line. So, they cancel each other. The forces F₁ and F₂ have equal magnitude but act in opposite directions on two sides of the coil. So, the forces F₁ and F₂ form a couple that tries to rotate the coil about its axis in an anti-clockwise direction.

Moment of the couple or Torque

τ = magnitude of either of the force × perpendicular distance between them

  = B I L × b cos θ
  = B I A cos θ  ( l × b = A = area of coil )

Hence, for acting on a coil in a magnetic field, τ = B I A cos θ If the coil possesses N turns, then the torque is given by,

τ = N B I A cos θ

or, τ = B I N A cos θ  …………………..(5)

Special cases

1. When θ = 0°, then,
     τ = B I N A
     Hence torque acting on the coil is maximum when plane of coil is parallel to the field.

2. When θ = 90°, then,

  τ = B I N A cos 90°

    = 0

Hence, no torque acts on the coil when the plane of the coil is perpendicular to the magnetic field

Moving coil galvanometer

A moving coil galvanometer is an electrical device that detects the flow of charge in an electrical circuit.

Principle: When a coil is placed in a magnetic field, it experiences a torque due to the forces acting on the coil.

A moving coil galvanometer consists of a coil containing a large number of turns of fine, insulated copper wire. The coil is wound over a light, non-magnetic metallic frame, which may be circular or rectangular in shape. It is suspended froma movable torsion head H by means of a fine phosphor-bronze wire. The lower end of the coil is connected to a fine spiral spring s’ of quartz. N and S are the pole pieces of a strong permanent magnet which surrounds coil. A soft iron core within the coil that the coil can rotate freely without touching the core. measurement. The torsion head is connected terminal T₁ and the spring is connected to terminal T₂.

In order to have a uniform scale of measurement, all positions of the coil during the movement must remain parallel to the magnetic field lines. This is achieved by using concave cylindrical pole pieces and a soft iron core. The magnetic field lines tending to concentrate into the soft iron core almost coincide with radii of the pole pieces that is why such a field is called the radial magnetic field.

Theory:

Let
N = total number of turns in the coil
A = Area of the coil
B = magnetic field induction of the radial magnetic field
I = current through coil

Then, the maximum deflecting torque τd is given by,
  τd = BINA

As the coil gets deflected, the suspension wire is twisted and a restoring toque is developed in coil. If k is the restoring torque per unit twist of the suspension wire, then the restoring torque for the deflection φ is given by

  Tr = kφ

So, I ∝ φ

Thus, deflection of the coil is directly proportional to the current flowing through it. Hence, we use a linear scale in the galvanometer to detect the current flowing in the circuit.

Sensitivity of a galvanometer

A galvanometer is said to be sensitive if a small amount of current flowing through its coil produces a large deflection in it. A galvanometer can be converted into an ammeter or a voltmeter so it has two types of sensitivity.

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

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

Hall effect:

When a magnetic field is applied to a current-carrying conductor, a voltage is developed across the specimen in the direction perpendicular to both the current and magnetic field. This effect is called as Hall effect. The transverse voltage produced in this effect is called as Hall voltage. It is denoted by V_H

As shown in the figure, a current is passing through a flat strip of metal in a given direction. A magnetic field B is applied at right angle to the strip due to which the strip experiences a force, which in turn is due to the force experienced by the charge carriers (free electrons for the metal). The magnetic force acting on the charge carriers having the magnitude e each moving with a drift velocity vₑ is Bev in the direction from bottom to top. According to Fleming’s left-hand rule, free electrons experience a force in the upward direction and the free electrons accumulate at the upper side of the strip, leaving a positive charge at its lower side, which makes the lower side at a positive (higher) potential and the upper side at a negative potential. In this way, a potential difference is set up. This particular potential difference between these two sides of the strip, which opposes further flow of charges from the lower to the upper side, is called the Hall potential difference or Hall voltage V_H

The electric force eE of free electrons to the upper face is given by

Biot And Savart Law (Laplace Law)

Consider a conductor XY through which current I is flowing. Due to it, a magnetic field is produced around it. To find the magnetic field produced at P, let us consider an element AB of length dl which makes an angle θ with the line r joining P and the Centre of the element dl as shown in the figure.

Let the distance of P from dl be r. According to Biot and Savart law, the magnetic field produced by the element at that point P is:
a) Directly proportional to magnitude of current passing through the conductor.
  i.e. dB ∝ I ——————(1)
b) Directly proportional to the length of the element.
  i.e. dB ∝ dL ——————(2)
c) Directly proportional to the sine of the angle between the conductor and the line joining r.
  i.e. dB ∝ sinθ ——————(3)
d) Inversely proportional to the square of the line joining between centre of the element and the point P.
  i.e.

Direction of dB:
The direction of the magnetic field at P can be determined by using the right-hand thumb rule. It follows that the direction of dB at P will be perpendicular to the plane containing. dl⃗ and .r⃗ 

Application of Biot-Savart law

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

Consider a circular coil of radius r having its center at O. When current I flows through the coil, a magnetic field is produced. To find the magnetic field at the center of the O of the coil, consider an element AB of length dl, which makes an angle θ = 90° with the line OC of length r.

According to Biot and Savart’s law, the magnetic field dB produced by the element at the centre of a circular coil is given by

To find out the magnetic field strength produced by whole coil, we have to integrate equation (1) along the circumference from 0 to 2πr.

i.e.

If the coil has N turns, each turn carrying current I, then the contributions of all the turns are added.

Direction of magnetic field

The direction of the magnetic field is perpendicular to the plane of the coil, and for the direction of current as shown in the figure, the magnetic field is directed inwards at the center of the coil.The magnetic lines of force are circular near the wire but practically straight near the centre of the coil.

The direction of the magnetic field at the centre of the coil is determined by using the right-hand fist rule.

2. Magnetic field on the axis of current-carrying circular coil

Suppose a narrow circular coil of radius ‘a’ in which current ‘I’ is flowing. Consider a point P on its axis at a distance x from the centre O of the coil.

According to Biot and Savart law, the magnetic flux density dB at a point P due to a small element dl of the coil is

Where r is the distance of the point P from the element dl. Here, the angle between dl and r is 90°. Therefore,

Suppose the line r makes an angle β with the axis. Thus, the small field dB has two components:

• dB cosβ (perpendicular to the axis)
• dB sinβ (along the axis)

If we suppose another diametrically opposite element dl, then the components perpendicular to the axis cancel each other, while the components along the axis add up. Hence, the total magnetic field due to one turn of a narrow coil is

If the coil contains N turns, then

This is the required expression.
Case I: when point P is at the center of the coil, $x=0$Then equation (3) becomes

This is the expression for the magnetic field at the centre of the current-carrying circular coil.

Case II: when point P is far away from the centre of the coil, x≫ a . So,

 a²+x²≈x²
Therefore, 

  3. Magnetic field on the axis of a current-carrying long solenoid

A long succession of narrow circular coils having a uniform number of turns per unit length is called a solenoid. In a solenoid, each loop can be considered as a circular coil.

Suppose, a long solenoid of radius a having n number of turns per unit length n and I be the current flowing through each turn of the solenoid. There is a point P on the axis of the solenoid at a distance r from the current-carrying element AB of the solenoid, having length dx. At this point, we have to find the magnetic flux density B due the whole solenoid. Referring to the figure, we have

PB = r    and   ∠PAB = β

Here, the total number of turns in the coil of length AB is

N=n×dx

Now, the magnetic flux density at P due to the length AB of N turns is given by

In triangle ΔADP, we have

If two ends of the solenoid subtend the angles β1​ and β2​ at point $P$, the total magnetic field at point $P$ due to the solenoid is

This is the required expression for the magnetic field due to a solenoid of finite length.

Special case:

If the point $P$ lies inside a solenoid of infinite length, then β1=0∘,β2=180∘

This is the expression for the magnetic field inside an infinitely long solenoid. The direction of magnetic field is given by the right-hand fist rule.

4. magnetic field at a point due to a long straight current carrying conductor

Consider a straight long conductor XY carrying current I. Let P be the point at a distance a from the conductor where magnetic flux density is to be determined.

Consider a small element of length dl at a distance r from P. According to Biot–Savart law, magnetic field at P due to a small element dl is

Direction of magnetic field:

The direction of the magnetic field is given by the right-hand thumb rule (right-hand grip rule).

According to this rule, when we grip the current-carrying conductor with our right hand such that the thumb shows the direction of current, then the curled fingers show the direction of the magnetic field produced by the current-carrying conductor.

Force between two current-carrying parallel conductors:

A current carrying conductor sets up a magnetic field around it. If such two conductors are kept very close to each other, then the force acts on each wire due to the magnetic field of the other. Therefore, they are attracted or repelled depending on the direction of the current. If the current flow is in the same direction, there is an attractive force between them, and if the current flow is in the opposite direction, there is a repulsive force between them.

Let us suppose two long and parallel conductors X and Y carrying currents I1, ​ and I2​, respectively in the same direction in air. Suppose r is the separation between these two parallel conductors. Therefore, the magnetic field B produced on the conductor Y due to the current flowing through the conductor X is

Similarly, the conductor $X$ experiences the force per unit length given by

The direction of the forces F_XY​  and F_YX​ is towards each other. Hence attractive force acts between two parallel conductors carrying like current.

If r=1 m, and I₁=I₂=1 A​

Definition of one Ampere:

One ampere electric current is defined in terms of force between two conductors:

“One ampere is that electric current which flows in each of the two infinitely long parallel wires of negligibly small area of cross-section separated by a distance of 1 m in air (vacuum) produces a force per unit length between the wires of 2×10−⁷ Nm−¹

Ampere’s Theorem (Ampere’s Circuital law):

Statement:
“The line integral of magnetic flux density $B$ over a closed path is numerically equal to μ0\mu_0μ0​ times the total current enclosed by the closed path.” Mathematically,

Proof:

Suppose a long straight wire in which current $I$ is flowing. The magnetic field $B$ due to this wire at a normal distance $r$ is

Suppose a circular closed path of radius r has the wire at its centre and perpendicular to its plane. Then the magnetic field $B$ is parallel to a small element $dl$ of the path. The line integral of the magnetic flux density B around the closed path is

Then the magnetic field B is parallel to a small element dl of the path. The line integral of the magnetic flux density B around the closed path is

Therefore,

Applications of Ampere’s theorem

1. Magnetic field due to a straight current-carrying conductor

Consider a long straight conductor carrying current I in the direction as shown in figure. It is desired to find out the magnetic field at a point P at a perpendicular distance from the conductor. The magnetic lines of force are concentric circles centered at the conductor.

We choose the circle of radius r as the closed path. The magnitude of B is same everywhere on this closed path. The angle between dl and B is 0∘ everywhere in this path. Thus, applying Ampere’s circuital law to this closed path, we get

Magnetic Field due to current long  carriying solenoid.

Suppose a long solenoid having n number of turns per unit length and carrying current I in the clockwise direction as shown in the figure. The magnetic field outside the solenoid is very small or almost zero, and the field inside is uniform. The direction of magnetic field B is along the axis of the solenoid.

To calculate the magnetic field B inside the solenoid, we suppose a rectangular closed path PQRSP as shown in the figure. Let PQ = x. Then the number of turns enclosed by the closed path is

N=nx

Thus, current enclosed by the closed path is

I′=NI=n I x………………(1)

The line integral of magnetic field B around the closed path is given by

B=μ₀​nI…(4)

This is the required expression for the magnetic field inside the current-carrying solenoid.

3. Magnetic field due to a toroid

A toroid is a solenoid bent into the form of a ring. The magnetic field outside the circular toroid is zero. Suppose a closed loop M is indicated by the broken line as shown in the figure.

The magnetic flux density B must be the same everywhere on this closed path. Therefore, the line integral of magnetic field intensity B due to the toroid is

Magnetic field Numerical

1) A straight horizontal rod of length 20 cm and mass 30 gm is placed in a uniform magnetic field perpendicular to the rod, if a current of 2A through the rod makes its sell supporting in the magnetic field, calculate the magnetic field. (Ans: 0.75 T)

2) A coil consisting of 100 circular loops with radius 60 cm carries a current of 5A. Find the magnetic field at a point along the axis of the coil, 80cm from the Centre.
(μ0=4π×10−⁷ Tm/A) (Ans: 1.13×10⁻⁴ T)

Given data:

B=1.13×10⁻⁴ T

3) A straight conductor of length 5 cm carries a current of 1.5 A. The conductor experiences a magnetic force of 4.5×10⁻3     when it is placed in a magnetic field of 0.9 N. What angle does the conductor make with the magnetic field? (Ans: 42°)

4) A straight conductor of length 15 cm is moving with a uniform speed of 10 m/s, making an angle of 30° with a uniform magnetic field of 10⁻⁴ Tesla. Calculate the emf induced across the length. (Ans: 7.5×10⁻⁵ V)

E=1.5×10−⁴×0.5=7.5×10⁻⁵   V

5) A circular coil consists of 100 turns and has a mean diameter of 20 cm. It carries a current of 5 A. Find the magnetic field at a point on the axis of the coil, 15 cm from its center. (μ0=4π×10⁻7 Tm/A)

Compiled by Er. Basant Kumar yadav