Revision: Magnetic Effects of Current and Magnetism CUET (UG) Magnetic Effects of Current and Magnetism

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.

A device used to accelerate positively charged particles (α particles, deutrons, protons, etc.) to acquire enough energy to carry out nuclear disintegration is called a cyclotron.

The turning effect experienced by a current-carrying loop placed in a uniform magnetic field, which forms the working principle of a moving coil galvanometer (MCG), is called torque on a current loop.

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 temperature above which a ferromagnetic substance becomes paramagnetic is called curie temperature. 

A line that joins all the places on earth, having the same angle of declination is called an isogonic line.

A line which joins all the places on earth, having zero angle of declination is called agonic line.

The path in a magnetic field in which a unit north pole tends to move when allowed to do so is known as magnetic field lines of force.

A line joining all the places on globe, having same angle of dip or inclination is called isoclinic line.

Substances which when placed in a magnetic field are feebly magnetised in a direction opposite to that of the magnetising field are called diamagnetic substances.

Substances which when placed in a magnetising field are strongly magnetised in the direction of the magnetising field are called ferromagnetic substances.

Substances which when placed in a magnetic field are feebly magnetised in the direction of the magnetising field are called paramagnetic substances.

Substances which at room temperature retain their ferromagnetic property for a long period of time are called permanent magnets.

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

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

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

mv = p = q BR

f_a​ = f_c​

Final energy in cyclotron: proportional to \[\ R_{exit}^2\]

\[F=BIL\sin\theta\]

Vector Form:

\[\vec{F}=I(\vec{L}\times\vec{B})\]

Special Cases:

• θ = 90^∘ → F = BILF 
• θ = 0^∘ → F = 0

Centripetal force provided by magnetic force: \[\frac{mv^2}{r}=qvB\]

Angular speed = \[\omega=\frac{qB}{m}\]

Period of circular motion = \[T=\frac{2\pi m}{qB}\]

Frequency = \[f=\frac{qB}{2\pi m}\]

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

Per unit length:

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

Force acts along the line joining the wires

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

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

\[B=\frac{\mu_0}{4\pi}\cdot\frac{m}{d^3}\]

\[B=\frac{\mu_0}{4\pi}\cdot\frac{2m}{d^3}\]

\[\tau=MB\sin\theta\]

Vector form: \[\vec{\tau}=\vec{M}\times\vec{B}\]

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.

If the thumb, forefinger and middle finger of the left hand are stretched mutually perpendicular to each other, and the forefinger points in the direction of the magnetic field and the middle finger points in the direction of the current, then the thumb gives the direction of the force acting on the conductor.

Tangent law states that, if a magnetic field ‘B’ is applied at right angles to the horizontal component of the earth's field B_H, the needle comes to equilibrium at an angle ‘$\theta ’$ to the magnetic meridian such that, tan θ = `B/B_H`.

For ferromagnetic substances above the Curie temperature, the magnetic susceptibility is inversely proportional to (T − T_C), where T_C is the Curie temperature. Mathematically,

On heating beyond the Curie temperature (T_C(iron) = 770 °C), ferromagnetic substances get converted into paramagnetic materials.

The magnetic susceptibility of a paramagnetic material varies inversely with its absolute temperature. Mathematically,

On cooling, paramagnetic substances get converted to ferromagnetic materials at the Curie temperature.

• Electric field accelerates the particle; magnetic field keeps it in circular orbit of constant frequency.
• Resonance: polarity of Ds reverses as ion crosses gap after each semicircle.
• Cannot accelerate neutrons (uncharged) or electrons (small mass, high velocity).
• Ion speed is limited.

• A current-carrying conductor placed in a magnetic field experiences a force when the direction of current is not parallel to the magnetic field.
• The direction of force reverses when the direction of current or the direction of magnetic field is reversed, and no force acts when current flows parallel to the magnetic field.

• Torque depends on current, magnetic field strength and area of the loop.
• For a given perimeter, a circular loop experiences maximum torque (maximum area).
• Forms the working principle of the moving coil galvanometer (MCG).

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

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

At equilibrium: \[NIAB=C\theta\]

• A current-carrying loop behaves like a magnetic dipole (bar magnet)
  
  
• Polarity Rule 
  Anticlockwise current → North pole (upper face)
  Clockwise current → South pole (lower face)

• A bar magnet behaves like a solenoid
• Both produce similar magnetic field patterns
• Solenoid Relation: M = NIA

• Relative permeability ranges: μ_r < 1 (as B is less than μ₀H); also 1 > μ_r > 0, μ < μ₀
• Diamagnetic: B < B₀​; B_m < B₀
• Magnetic susceptibility (χ): low and negative, ∣χ∣ ≈ 1; small, negative and temperature-independent, χ_m ∝ T₀
• Magnetic moment: very low (≈ 0)
• Intensity of magnetisation (I) vs H: I is small, negative, varies linearly with H (I and H in opposite direction, I is negative with respect to H)

• Relative permeability ranges:   μ_r ≫ 1, of the order of 10²; μ ≫ μ₀
• Diamagnetic: B ≫ B₀​; B_m ≫ B₀
• Magnetic susceptibility (χ): positive and high, χ ≈ 10²; very large, positive, temperature dependent, χ_m ∝ \[\frac {1}{T−T_C}\]​ (Curie–Weiss law)
• Magnetic moment: very high
• Intensity of magnetisation (I) vs H: I is very large, positive, varies non-linearly with H (I is in the direction of H, value of I is very high)

• Relative permeability ranges:  μ_r > 1 (as B is slightly greater than μ₀H); (1 + ε) ≥ μ_r > 1, μ > μ₀
• Diamagnetic: B < B₀​; B_m < B₀
• Magnetic susceptibility (χ): low and positive, χ ≈ 1; small, positive, varies inversely with temperature, χ_m ∝ \[\frac {1}{T}\]​ (Curie law)
• Magnetic moment: very low but not zero
• Intensity of magnetisation (I) vs H: I is small, positive, varies linearly with H (I and H in same direction, value of I is low)