Overview: Moving Charges and Magnetic Field

The region near a magnet, where a magnetic needle experiences a torque and rests in a definite direction, is called 'magnetic field'.

If a charge of 1 coulomb moving with a velocity of 1 metre per second perpendicular to a uniform magnetic field experiences a force of 1 newton, then the magnitude of the field is 1 tesla.

The direction in the magnetic field along which a current-carrying conductor does not experience any force is called the direction of the magnetic field.

If we stretch our right-hand palm such that the thumb points in the direction of the current (I) and the stretched fingers in the direction of the magnetic field \[\vec B\], then the force \[\vec F\] on the conductor will be perpendicular to the palm in the direction of pushing by the palm.

If the forefinger, the middle finger, and the thumb of the left hand are stretched at right angles to one another, such that the forefinger points in the direction of the magnetic field \[\vec B\] and the middle finger in the direction of the current I, then the thumb will point in the direction of the force \[\vec F\] on the conductor.

1 ampere is the current which when flowing in each of two infinitely-long parallel conductors 1 metre apart in vacuum, produces between them a force of exactly 2 × 10^-7 newton per metre of length.

F = Bq v sin θ

or,

B = \[\frac{F}{qv\sin\theta}\]

Units: 1 T = 1 NA^-1 m^-1 = 1 Wb m^-2

Dimensions: [M T^-2A^-1].

\[\vec F\] = q(\[\vec E\] + \[\vec v\] × \[\vec B\])

\[\frac{F}{L}=\frac{\mu_0}{2\pi}\frac{I_1I_2}{r}\]

OR

\[F=\frac{\mu_0I_aI_bL}{2\pi d}\]

Statement

The magnetic field produced at a point due to a small current element is directly proportional to the current through the element, the length of the element, and the sine of the angle between the element and the line joining it to the point, and inversely proportional to the square of the distance of the point from the element.

Proof

Consider a conductor of arbitrary shape carrying a current I. Let dl be a small current element of the conductor and r the distance of this element from a point P.

According to experimental observations:

1. The magnetic field dB at point P is proportional to the current I:
   dB ∝ I

2. It is proportional to the length of the current element dl:
   dB ∝ dl

3. It is proportional to sin⁡ θ, where θ is the angle between dl and the line joining the element to point P:
   dB ∝ sin⁡θ

4. It is inversely proportional to the square of the distance r:
   dB ∝ \[\frac {1}{r^2}\]​

Combining all these relations: dB ∝ \[\frac{Idl\sin\theta}{r^2}\]​

For air or vacuum, this proportionality is written as:

dB = \[\frac{\mu_{0}}{4\pi}\frac{Idl\sin\theta}{r^{2}}\]​

The direction of dB is perpendicular to the plane containing dl and the position vector \[\vec r\]. Hence, in vector form:

d\[\vec B\] = \[\frac{\mu_{0}}{4\pi}\frac{I(d\vec l\times \vec r)}{r^{3}}\]​

The magnetic field due to the entire conductor is obtained by integrating over its length:

\[\vec B\] = ∫ d\[\vec B\] = \[\frac{\mu_0I}{4\pi}\int\frac{d\vec l\times\mathbf{\vec r}}{r^3}\]

Conclusion

Biot–Savart’s law gives the magnitude and direction of the magnetic field produced by a current-carrying conductor. It shows that the magnetic field depends on the current element, its orientation, and its distance from the point, and that it forms the basis for calculating magnetic fields due to finite-sized conductors.

Biot-Savart Law in Terms of Current Density j

\[j=\frac{I}{A}=\frac{Idl}{Adl}=\frac{Idl}{dV},\]

where dV is the volume of current element.

\[\therefore\] Idl = j dV

d\[\vec B\] = \[\frac{\mu_0}{4\pi}\frac{\vec j\times\vec r}{r^3}\]

Units: kg m s² A^-2

Dimensions: [M L T²A^-2].

Statement:
The line integral of the magnetic field B around any closed path in free space is equal to μ0​ times the net steady current enclosed by the path.

\[\oint\vec{B}\cdot d\vec{l}=\mu_0I\]

Explanation/Proof (for a long straight conductor):
Consider a long, straight conductor carrying a steady current I.
The magnetic field at a distance r from the conductor is

B = \[\frac{\mu_0I}{2\pi r}\]​

The field lines are circular, and B is tangential and constant in magnitude along a circular path of radius r.
Hence,

\[\oint\vec{B}\cdot d\vec{l}=B\oint dl=B(2\pi r)\]

Substituting the value of B,

\[\oint\vec{B}\cdot d\vec{l}=\mu_0I\]

Thus, Ampere’s circuital law is verified.

Conclusion:

• Ampere’s circuital law is valid for steady currents and time-independent magnetic fields.
• It is applicable to any closed path.
• It is especially useful for conductors with high symmetry, where calculating the magnetic field is easier.
• It is analogous to Gauss’s law in electrostatics.

• Current produces magnetism: Oersted showed that a current-carrying conductor produces a magnetic field around it, as indicated by the deflection of a magnetic needle.
• Direction and strength: Reversing the direction of the current reverses the needle’s deflection, and increasing the current or reducing the distance increases the deflection.
• Moving charges and force: Since current is the flow of moving charges, the experiment shows that moving charges create magnetic fields and exert forces.

• Coulomb’s law describes the electric field due to charges, while Biot–Savart law describes the magnetic field due to a current.
• In both cases, the field strength decreases with the square of the distance.
• Biot–Savart law is the magnetic counterpart of Coulomb’s law.
• The electric field depends only on distance, but the magnetic field also depends on the angle of the current element.
• The electric field acts along the line joining the source and the point, while the magnetic field acts perpendicular to it.

• Right-hand palm rule: Thumb shows current direction; perpendicular from the palm gives magnetic field direction.
• Right-hand thumb rule: Thumb points along current; curled fingers show magnetic field direction.
• Maxwell’s right-hand screw rule: Direction of screw motion gives current; direction of rotation gives magnetic field.

• The magnetic field at a point P at a distance r from a finite straight current-carrying conductor is
  B = \[\frac{\mu_0I}{4\pi r}(\sin\phi_1+\sin\phi_2)\]
  where ϕ₁ and ϕ₂​ are the angles subtended by the conductor at P.
• For special cases of a straight conductor:
  Infinite length: B = \[\frac{\mu_0I}{2\pi r}\]
  Point near one end: B = \[\frac{\mu_0I}{4\pi r}\]​
  The magnetic field is directly proportional to current I and inversely proportional to distance r.
• The magnetic field at a point on the axis of a circular loop of radius a, carrying current I, at a distance x from the centre is
  B = \[\frac{\mu_0Ia^2}{2(a^2+x^2)^{3/2}}\]​
  For a coil of N turns, the field is multiplied by N.
• At the centre of a circular current-carrying loop, the magnetic field is
  B = \[\frac{\mu_0I}{2a}\]​
  For a coil of N turns,
  B = \[\frac{\mu_0NI}{2a}\]​
• The direction of the magnetic field is given by the right-hand rule. At the centre of a circular coil, magnetic field lines are nearly straight, parallel, and perpendicular to the plane of the coil, indicating a nearly uniform magnetic field.

• For a long straight current-carrying wire, the magnetic field at a distance r is
  B = \[\frac{\mu_0I}{2\pi r}\]​
  The field lines are concentric circles and B ∝ \[\frac {1}{r}\]
• Inside a long, straight solenoid, the magnetic field is uniform and given by
  B = μ₀nI
  where n is the number of turns per unit length; the field outside the solenoid is nearly zero.
• The magnetic field at the end of a long solenoid is half of that at the centre:
  B_end = \[\frac {1}{2}\]μ₀nI
• For a solenoid with a magnetic core of relative permeability μ_r, the field becomes
  B = μ₀μ_rnI
• In a toroidal (endless) solenoid, the magnetic field exists only inside the core and is
  B = \[\frac{\mu_0NI}{2\pi a}\] = μ₀nI
  The field is zero outside the toroid and varies with the radial distance a.

• A charged particle q moving with velocity v in a uniform magnetic field B experiences a force
  F = Bqv sin⁡θ or \[\vec F\] = q(\[\vec v\] × \[\vec B\])
  where θ is the angle between \[\vec v\] and \[\vec B\].
• The magnetic force acts perpendicular to both the velocity \[\vec v\] and the magnetic field \[\vec B\]; its direction is given by the right-hand screw rule for positive charges and is opposite for negative charges.
• Fleming’s left-hand rule can also be used: forefinger → magnetic field, middle finger → current (direction of positive charge motion), and thumb → direction of magnetic force.

• Magnetic force on a moving charge: A charged particle of charge q moving with velocity v in a uniform magnetic field B experiences a force
  F = qv B sin⁡ θ, 
  where θ is the angle between \[\vec V\]  and \[\vec B\].
• Case θ = 0^∘(velocity parallel to field): When the particle moves parallel to the magnetic field, the magnetic force is zero, and the particle moves in a straight line without deviation.
• Case θ = 90^∘ (velocity perpendicular to field): When the velocity is perpendicular to the magnetic field, the particle experiences a constant force perpendicular to its velocity and moves in a circular path at a constant speed.
• Radius of circular path: In perpendicular entry, the magnetic force provides the centripetal force, giving the radius:
  r = \[\frac {mv}{qB}\]​
  Hence, the radius is proportional to the momentum of the particle.
• Time period and frequency: The time period of the revolution is
  T = \[\frac {2πm}{qB}\]​
  and is independent of the speed of the particle. The angular frequency is
  ω = \[\frac {qB}{m}\].
• Case 0^∘ < θ < 90^∘ (oblique entry): When the particle enters at an angle, it follows a helical path formed by the combination of circular motion (due to perpendicular component) and linear motion (due to parallel component).
• Speed and energy remain constant: For all angles between \[\vec v\] and \[\vec B\], the speed, kinetic energy, and time period remain unchanged, but the direction of momentum changes when θ ≠ 0^∘.

• Principle: A cyclotron accelerates charged particles using an alternating electric field and a magnetic field to bring them repeatedly to the accelerating gap.
• Construction: It has two D-shaped hollow electrodes (dees) placed in a uniform magnetic field perpendicular to their plane.
• Motion of Particle: Inside the dees, the particle moves in a circular path with radius
  r = \[\frac {mv}{qB}\]​
  and gains energy only in the gap.
• Resonance Condition: For continuous acceleration, the frequency of the applied voltage equals the particle’s cyclotron frequency:
  ν = \[\frac {qB}{2πm}\]​
• Limitations: A cyclotron cannot accelerate neutral particles or electrons and is ineffective at very high (relativistic) speeds.

• Existence of Force: A current-carrying conductor placed in a uniform magnetic field experiences a force perpendicular to both the direction of current and the magnetic field.
• Direction of Force: The direction of force can be determined by Fleming’s Left-Hand Rule or Right-Hand Palm Rule.
• Expression for Force
  The magnitude of force acting on a straight conductor of length l is
  F = B I l sin⁡ θ
  where θ is the angle between the conductor and the magnetic field.
• Special Cases:
  When θ = 0^∘, the force is zero.
  When θ = 90^∘, the force is maximum, F_max⁡ = B I l.
• Vector Form: The magnetic force on a current-carrying conductor can be written as
  \[\vec F\] = I \[\vec l\] × \[\vec B\]