Revision: Magnetic Effects of Current and Magnetism >> Moving Charges and Magnetism Physics Science (English Medium) Class 12 CBSE

The force experienced by a moving charge in the presence of a magnetic field, which depends on charge q, velocity v and magnetic field B, and which is opposite in direction on a negative charge compared to a positive charge, is called the magnetic force.

Current passed through each of the two infinitely long parallel straight conductors kept at a distance of one meter apart in vacuum causes each conductor to experience a force of 2 × 10^-7 newton per meter length of the conductor.

A long solenoid is a coil whose length is much greater than its radius, producing a uniform magnetic field inside and nearly zero field outside.

The current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit current flows through it.  
Mathematically, it can be given by:

I_S = `(NBA)/k`

Where k is the couple per unit twist.

Current sensitivity is defined as the deflection e per unit current.

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

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 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.

A moving-coil galvanometer is a sensitive instrument used to detect and measure electric current, based on the principle that a current-carrying coil placed in a magnetic field experiences a deflecting torque proportional to the current.

The current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit current flows through it.

“The magnetic moment of a coil is equal to the maximum torque acting on a coil when placed in a uniform magnetic field of unit strength".

Units: NIA as A-m²

The voltage sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit voltage is applied across its coil.

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}\]

Magnetic force when v ∥ B: F = 0

Magnetic force when velocity is zero: ∣F_m∣ = 0

Maximum magnetic force (when v ⊥ B): F_max = qv B

F = qv B sin θ

F = IL × B

\[\vec{B}=\frac{\mu_0IR^2}{2(x^2+R^2)^{3/2}}\hat{i}\]

Where:

• I = current
• R = radius of loop
• x = distance from centre along axis
• μ0 = permeability of free space

B = μ₀​nI

Where:

• μ_0​ = permeability of free space
• n = number of turns per unit length
• I = current

\[F=\frac{\mu_0I_1I_2}{2\pi d}\times l\]

Per unit length:

\[\frac{F}{l}=\frac{\mu_0I_1I_2}{2\pi d}\]

Force acts along the line joining the wires

\[m=NIA\]

\[\tau=NIAB\sin\theta\]

Also written as:

\[\vec{\tau}=\vec{m}\times\vec{B}\]

\[\mathrm{V.S.}=\frac{\theta}{V}=\frac{NBA}{CR}\]

\[\mathrm{C.S.}=\frac{\theta}{I}=\frac{NBA}{C}\]

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

OR

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

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(\[\vec E\] + \[\vec v\] × \[\vec B\])

\[\frac{\phi}{I}=\frac{NAB}{c}\]

OR

\[\frac{\phi}{I}=\frac{NAB}{k}\]

τ = Ι Α B sin θ

Vector form:

\[\vec τ\] = \[\vec m\times\vec B\]

\[\frac{\phi}{V}=\frac{NAB}{cR}\]

OR

\[\frac{\phi}{V}=\frac{NAB}{kR}\]

If we stretch the index finger, middle finger and thumb of the left hand mutually perpendicular to each other such that the index finger points along the direction of the magnetic field and the middle finger along the direction of current (moving charge), then the thumb represents the direction of the force F experienced by the moving charge.

The magnitude of magnetic induction (dB) at a point due to a small element of current carrying conductor is:
(i) directly proportional to current (dB ∝ I),
(ii) directly proportional to length of element (dB ∝ dl),
(iii) directly proportional to sine of angle between element and line joining its centre to the point (dB ∝ sin θ),
(iv) inversely proportional to square of distance (dB ∝ 1/r²).

Applications

• Magnetic field at centre of circular coil.
• Magnetic field on axis of the coil.
• Magnetic field at a distance from a straight current-carrying conductor.

This law states that the line integral of magnetic field density (B) along an imaginary closed path is equal to the product of the current enclosed by the path and the permeability of the medium.

\[\oint\vec{B}.\overrightarrow{dl}=\mu_{0}I\]

The line integral of magnetic field of induction \[\vec B\] around any closed path in free space equals μ₀​ times the total current through the area bounded by the path.
∮\[\vec B\] ⋅ \[\vec d\]s = μ₀​I. The closed loop is called an Amperian loop; I is the net current enclosed.

Applications

• Magnetic field due to a long straight current-carrying wire.
• Magnetic field inside an ideal long straight solenoid.
• Magnetic induction along the axis of a toroid.

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.

Based on the torque on the current-carrying coil in the magnetic field: \[\tau=NIAB\]

Restoring torque: \[\tau=C\theta\]

At equilibrium: \[NIAB=C\theta\]

• 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 \[\vec l\] × \[\vec B\]

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• A moving-coil galvanometer is used to detect and measure small electric currents.
• It works on the principle that a current-carrying coil in a magnetic field experiences a torque.
• There are two types: suspended-coil (more sensitive) and pivoted-coil (Weston; more convenient).
• A radial magnetic field is used so that deflection is directly proportional to the current.
• The coil comes to rest when the deflecting torque equals the restoring torque.
• A shunt is connected in parallel to protect the galvanometer from high currents and to enable null-point measurements.

• A voltmeter is used to measure potential difference and is always connected in parallel across the points of measurement.
• An ideal voltmeter has infinite resistance so that it draws no current from the circuit.
• A galvanometer is converted into a voltmeter by connecting a high resistance in series with it.
• The resistance of a voltmeter is much greater than the resistance of the galvanometer.
• The higher the voltmeter's range, the greater its resistance; a lower-range voltmeter has lower resistance.

• An ammeter is used to measure electric current and is always connected in series in a circuit.
• An ideal ammeter has zero resistance, but a galvanometer has appreciable resistance and cannot be used directly as an ammeter.
• To convert a galvanometer into an ammeter, a low resistance shunt is connected in parallel with the galvanometer.
• The shunt allows most of the current to bypass the galvanometer, so only a small, safe current flows through the coil.
• The shunt's value depends on the ammeter's range and ensures that full-scale deflection corresponds to the desired maximum current.

• A current-carrying loop placed in a uniform magnetic field experiences a turning effect (torque).
• Equal and opposite forces act on opposite sides of the loop, forming a couple that tends to rotate the loop.
• The forces on the remaining sides cancel each other, so the net force on the loop is zero.
• The torque depends on the loop's orientation in the magnetic field and is zero at a particular position.
• This effect is the basic principle of galvanometers and electric motors.