Magnetic Field and Magnetic Force

What is a Magnetic Field?

The magnetic field can be defined in multiple ways depending on its situation. Generally, a magnetic field is defined as an invisible field around a magnetic object. A magnetic field is generally used to express the distribution of magnetic force around a magnetic object.

Magnetic fields are formed when the electric charge or electric current moves within the domain of the magnet. Here, the subatomic particles such as negatively charged electrons move around forming a magnetic field. The magnetic field can be formed either inside the atoms of magnetic objects or within the electrical wires or conductor. 

Magnetic Field Representation

The magnetic field can be represented in multiple ways. Mathematically, it is represented as a vector field which can be arranged as a different set on a grid. The second method to represent the magnetic field is by using the field lines. The collection of vectors are joined with lines. Here, the magnetic field never stops and meets each other.

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Magnetic Field Measurement and Unit

The measurement of the magnetic field includes the estimation of both its strength and direction. The measurement is important because each magnetic field is distinct from one another. The strength of the magnetic field can either be small and weak or large and strong. However, the term magnetic field denotes two unique but related fields which are represented by using the symbol H and B. The strength of the magnetic field is represented by H and is estimated in ampere per metre.

Simultaneously, the magnetic flux is denoted by B and is measured in Tesla.

Magnetic Force 

Magnetic force is an outcome of electromagnetic force and it arises due to the movement of charges. As we know, the moving charges encircle itself with a magnetic field. From this point of view, the magnetic force can be defined as a force that is caused due to the interactive magnetic field. It can either be a repulsive or attractive force. The magnetic force operating on a moving charge in the existence of magnetic fields.

A charge ‘x’ is moving with a velocity ‘v’ forming an angle ϴ with the field direction. Experimentally, we determined that a magnetic force operating the moving charge is represented by F\[^{\rightarrow }\]\[_{B}\] = q (V\[^{\rightarrow }\] ∗ B\[^{\rightarrow }\]) . It is known as Lorentz force law.

Key Features

• No magnetic force can act on a fixed charge present in the magnetic field.

• No magnetic force can operate on moving charge when it is moving in the direction of either antiparallel or parallel in the magnetic field. 

• The magnetic force functioning on moving charge is maximum when a charge is moving ⏊ ar to \[\vec{B}\].

As (V\[^{\rightarrow }\] ⊥ B\[^{\rightarrow }\]) and Ө = 90 degrees ⇒ F\[^{\rightarrow }\] \[_{B}\] is maximum 

No work can be performed by the magnetic force on a moving charge because F\[^{\rightarrow }\] \[_{B}\] is always ⊥ ar to B\[^{\rightarrow }\].

Hence, there is no variation in the K.E. of the charge. The direction of movement may change but there will not be any change in speed.

Magnetic Force on Current

Let us examine a line charge ‘’λ‘’ moving with a velocity ‘v’ as represented in the below figure. The amount of charge intersecting a point p in a timeslot of Δt is λ V Δt. The rate of flow of charge around point p is the line current represented as I.

I = λ V Δt/Δt = λV ; I\[^{\rightarrow }\]

λV\[^{\rightarrow }\] (In vector form) ⇒ A charge moving forward with velocity ‘v’ ( a discrete charge, line charge, volume charge, surface charge etc) is equal to the current “I”. The force functioning on a line charge moving with a velocity ‘v’ or the magnetic force operating on a  line current is represented by  

\[\vec{F}\] = (\[\vec{V}\] ∗ \[\vec{B}\])dq = \[\int\]\[_{L}\] ((\[\vec{V}\] ∗ \[\vec{B}\])λdl = \[\int\]\[_{L}\] (\[\vec{I}\] ∗ \[\vec{B}\] )dl = I \[\int\]\[_{L}\] (\[\vec{dl}\] ∗ \[\vec{B}\])    

Note: I is eliminated from the integral because the current which we are determining is steady currents. A conductor is holding a current ‘I’, and is present in a magnetic field.

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\[\vec{F}\] = I \[\int_{P}^{Q}\] \[\vec{dl}\] ∗ \[\vec{B}\].

If \[\vec{B}\] is uniform,

\[\vec{F}\] = I [[\[\int_{P}^{Q}\] \[\vec{dl}\] ∗ \[\vec{B}\]] ∗ \[\vec{B}\]]  = I[\[\vec{PQ}\] ∗ \[\vec{B}\]].

If it is a closed current-carrying conductor, carrying a constant current and is placed in a magnetic field \[\vec{B}\].

\[\vec{F}\] = I \[\oint\]\[_{L}\]\[\vec{dl}\] ∗ \[\vec{B}\]     

If \[\vec{B}\] is uniform, then the magnetic force functioning on it will always be equivalent to zero.

Magnetic Field of A Straight Line

A straight line current-carrying conductor is transferring a current ‘I’ as represented in the figure. With the help of Big- Savart's Law, we are now determining the magnetic field at a point p, which is placed at distance ‘r’ from the wire. 

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In the above figure, we can see OA = Z cos θ =  r/PA

R tan θ = Z

Dz = r sec² θ dθ

PH = r sec θ 

dB = μO IdZ sin ∠PAC / 4πr (PA)²

= μO I r sec² θ dθ cos θ / 4π (r² sec² θ) = μO I cos θ dθ / 4πr

= x [\[\bar{B}\] = μO I/ 4πr ∫\[^{\frac{\pi }{2}}\] cos θ dθ = μO I/4πr] x =  \[\vec{B}\]. = μO I/ 4 πr \[\int_{-\theta _{1}}^{\theta _{2}}\] cos θ dθ 

= μO I/ 4πr ( sin (θ₁) + 8n (θ₂))

Case 1: When an infinite wire is carrying a current I, the angle crossover at point B can be determined as θ₁ = θ₂ = π/2 ⇒ \[\vec{B}\]. = μO I/ 4πr (2) =  μO I/ 2πr

Case 2: If the length of the wire extends to infinity and the point P is placed on a  perpendicular passing through one end of the below semi-infinite wire.

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B = μO I/ 4πr [ 8n0₂ + 0] = μO I / 4π  θ₁ = 0°, θ₂ = 90°

Case 3: If the point p is placed on a ar bisector of length 2L.

B = μO I/4πr (28n θ) = μO I8nθ

Or

\[\vec{B}\] = μO I/ 2πr L / \[\sqrt{L^{2} + r^{2}}\] 

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Magnetic Field of a Solenoid

A coil of wire which is drawn to form a strong magnetic field within the coil is known as a solenoid. Enfolding the same wire repeatedly around the cylinder forms a strong magnetic field when an electric current is passed through it. The symbol ‘N’ is used to represent the total number of turns the solenoid retains. The magnetic field will be stronger with the maximum number of loops.

A solenoid is a kind of electromagnet whose aim is to produce a controlled magnetic field. The solenoid will be characterized as an inductor if the solenoid aims to slow down changes in the electric current. 

The magnetic field of a solenoid formula is derived by:

B = μO IN/L

In the above formula, 

N represents the total number of turns in a solenoid

I represent the current of the coil

L represents the length of the coil

Note: The magnetic field in the coil is always equivalent to the applied current and the total number of turns per unit length.

Magnetic Field Due to the Current-Carrying Circular Loop

The magnetic field forms the shape of a circular loop at each point of the current-carrying circular loop. The magnetic field lines become straight and perpendicular to the plane of the coil at the centre of the circular wire. 

Right- Hand thumb rule is used to represent the direction of the magnetic field.

• The magnetic field lines are always straight in the middle of the circular loop.

• Each part of the circular loop carrying current produces magnetic field lines in a similar direction within the loop.

• The direction of the magnetic field at the middle of the circular coil is perpendicular to the plane of the coil. i.e. around the axis of the coil.

Magnetic Field Change on the Axial Distance

The Magnetic field is maximum at the centre but when we are moving to each side of the circular loop along the Z-axis, we can see the field is changing non-linearly at two different points together with the axis, the second derivative B is disappearing and fetching zero.

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The change of \[\vec{B}\] with Z as constant at these two points 🡺 . The change of \[\vec{B}\] is linear at these two points. 

These two points are defined as ‘ the point of inflection' of the graph’.