Revision: 12th Std >> Magnetic Fields Due to Electric Current MAH-MHT CET (PCM/PCB) Magnetic Fields Due to Electric Current

The phenomenon by virtue of which an electric current in a conductor produces a magnetic field around it is called the magnetic effect of electric current.

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.

When both electric and magnetic fields act on a charge, the total force is called the Lorentz force.

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 mutual force experienced by two current-carrying wires — attractive if currents are in the same direction and repulsive if currents are in opposite directions — is called the force between two current-carrying wires.

The force experienced by a current-carrying conductor placed in a uniform magnetic field is called the force on a current-carrying conductor.

The net force experienced by a closed circuit placed in a uniform magnetic field, which is always zero, is called the force on a closed circuit in a uniform magnetic field.

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 energy possessed by a magnetic dipole freely suspended in a magnetic field due to its orientation in the field is called its magnetic potential energy.

The lines of constant magnitude of magnetic field around a current-carrying wire which form concentric circles and are tangential at every point to the direction of field are called magnetic field lines.

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.

An anchor ring (torus) around which a large number of turns of metallic wire are wound, forming an endless solenoid, is called a toroid.

An insulated long wire closely wound in the form of a helix, whose length is very large compared to its diameter, is called a solenoid.

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

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

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

Vector Form: \[\vec F\] = q(\[\vec v\] × \[\vec B\])

Magnitude Form: F = qv B sin θ

Where:

• q = charge on the particle
• v = speed of the particle
• B = magnetic field strength
• θ = angle between \[\vec v\] and \[\vec B\]

mv = p = q BR

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

f_a​ = f_c​

m = IA (or m = NIA)

Combining the four dependencies: dB ∝ \[\frac {I dl sin θ}{r^2}\]​

Introducing the constant of proportionality \[\frac {μ_0}{4π}\]​​:

\[dB=\frac{\mu_0}{4\pi}\cdot\frac{Idl\sin\theta}{r^2}\]

\[{d\vec{B}=\frac{\mu_0}{4\pi}\cdot\frac{Id\vec{l}\times\hat{r}}{r^2}=\frac{\mu_0}{4\pi}\cdot\frac{Id\vec{l}\times\vec{r}}{r^3}}\]

For a finite conductor, integrate over the entire length:

\[{\vec{B}=\frac{\mu_0I}{4\pi}\int\frac{d\vec{l}\times\hat{r}}{r^2}}\]

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

Where:

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

F = IL × B

Inside (r < R): B_in = \[\frac {μ_0Ir}{2πR^2}\]

At surface (maximum): B_s = \[\frac {μ_0I}{2πR}\]

Outside (r > R): B_out = \[\frac {μ_0I}{2πr}\]

\[B=\mu_0nI\]

\[n=\frac{N}{L}\] (number of turns per unit length)

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.

If we stretch our right hand such that the fingers point towards the point at which magnetic field is required while the thumb is in the direction of current, then the normal to the palm will show the direction of the magnetic field.

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.

Two parallel current-carrying conductors with currents in the same direction attract each other; with currents in opposite directions, they repel.

Statement

The line integral \[\oint\vec{B}\cdot d\vec{l}\] taken around any closed loop equals μ₀ times the net steady current passing through the loop.

Proof (for a long straight wire)

• Consider an infinitely long straight wire carrying current I.

• By Biot–Savart law, field at distance r:
  B = \[\frac{\mu_0I}{2\pi r}\]

• Choose a circular Amperian loop of radius r, concentric with the wire.

• By symmetry, B is constant in magnitude and tangential (parallel to \[d\vec l\]) everywhere:
  \[\oint\vec{B}\cdot d\vec{l}=B\oint dl\] = B(2πr)

• Substituting B:
  \[\oint\vec{B}\cdot d\vec{l}=\frac{\mu_0I}{2\pi r}(2\pi r)\] = μ₀​I

Conclusion

\[\oint\vec{B}\cdot d\vec{l}=\mu_0I\]
The result is independent of the loop's radius, confirming the law's validity.

The toroid is a solenoid bent into the shape of a hollow doughnut.

According to Ampere's circuital law.

`phivecB.vec(dL) = mu_0I`

Here current 'I' flow through the ring as many times as there are the N no. of turns.

∴ `phivecB.vec(dL) = mu_0NI` ......(1)

Now, B and dL are in the same direction.

∴ `phivecB.vec(dL) = BphidL`

∴ `phivecB.vec(dL) = B.(2pir)` .....(2)

From (1) and (2),

`mu_0NI = B.(2pir)`

∴ B = `(mu_0NI)/(2pir)`

• 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.

• 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).

• Direction given by right-hand thumb rule; for a loop, B at centre and M are parallel.
• Magnetic moment of a straight current-carrying wire = 0.
• Magnetic moment of a toroid = 0.
• Dipole moment direction: S → N (inside magnet field taken N → S).

• Magnetic field has the same magnitude at every point on a circle of radius r — cylindrical symmetry.
• Field direction is tangential to this circle.
• Even for an infinite wire, field at a non-zero distance is not infinite.

• B inside is independent of length and diameter and is uniform across the cross-section.
• B at the ends = ½ × B at the centre.
• B outside the solenoid is zero.
• Field pattern is similar to that of a bar magnet.
• B inside the toroid is independent of r, provided turns per unit length remain same.
• B outside the toroid is zero; field is confined to the core on which winding is made.