Revision: Magnetic Fields Due to Electric Current Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

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.

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 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 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 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.

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.

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

A solenoid is a long, closely wound helical coil of wire that produces a nearly uniform magnetic field inside it when current flows through it.

A toroid is a solenoid bent into a closed circular (ring-shaped) form.

When a charged particle moves perpendicular to a uniform magnetic field, it undergoes uniform circular motion.

The magnetic dipole moment of a current-carrying coil is defined as the product of the number of turns, the current, and the area of the coil.

μ = N I A

When a charged particle enters a uniform magnetic field with velocity having both perpendicular and parallel components to the field, it moves in a helical path.

A moving coil galvanometer is an instrument used to detect and measure small electric currents based on the torque acting on a current-carrying coil placed in a magnetic field.

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

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

F = qv B sin θ

Magnetic force when v ∥ B: F = 0

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

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

mv = p = q BR

f_a​ = f_c​

M_net​ = \[\sqrt{M_1^2+M_2^2+2M_1M_2\cos\theta}\],

tan α = \[\frac{M_2\sin\theta}{M_1+M_2\cos\theta}\].

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 = \[\frac{\mu_0I}{2R}\]

Magnetic Field at Centre of a Coil (N turns):

B = \[\frac{\mu_0NI}{2R}\]

\[B_z=\frac{\mu_0IR^2}{2(R^2+z^2)^{3/2}}\]

Where:

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

Magnetic Field Inside a Long Solenoid:

B = μ_0​ni

Magnetic Field Outside an Ideal Solenoid:

B = 0

B = \[\frac{\mu_0Ni}{2\pi R}\]

F_m​ = q(v × B)

τ = μB sin θ

or in vector form,

τ = μ × B

τ = NIAB sin θ

Where:

• N = number of turns
• I = current
• A = area of the loop
• B = magnetic field
• θ = angle between the magnetic field and normal to loop

U = −μ⋅B

Scalar Form: U = −μB cos θ

U_min ​= −μB

U_max​ = +μB

F = I ∫ dl × B

p = qBR

B = \[\frac{\mu_0I}{2\pi d}\]

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

K.E. = \[\frac{1}{2}\mathrm{mv}^{2}=\frac{q^{2}B^{2}R_{exit}^{2}}{2m}\]

τ = μB sin θ

τ = NIAB

Deflection Relation:

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

F = I ∮ dl × B

F = IL × B

\[f_c=\frac{1}{T}=\frac{qB}{2\pi m}\]

B = \[\frac{\mu_0I\theta}{4\pi r}\]

For θ = 2π,

B = \[\frac{\mu_0I}{2r}\]

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.

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.

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)`

Statement

The line integral of the magnetic field around any closed loop is equal to μ₀ times the net current enclosed by the loop.

∮ B ⋅ dl = μ₀I_enc​

where

• B = magnetic field
• dl = small element of the closed loop
• I_enc = net current enclosed
• μ_0​ = permeability of free space

Explanation / Proof 

Consider a long straight wire carrying current I.

Due to cylindrical symmetry:

• Magnetic field B is tangential to a circular path around the wire.
• Magnitude of B is the same at all points on a circle of radius r.

Choose a circular Amperian loop of radius r.

Since B and dl are parallel:

∮ B ⋅ dl = ∮ B dl

=B∮dl

=B(2πr)

By Ampere’s Law:

B(2πr) = μ₀I

B = \[\frac{\mu_0I}{2\pi r}\]​

This matches the magnetic field obtained earlier.

Conclusion

Hence,

∮ B ⋅ dl = μ₀I_enc​

is verified and is known as Ampere’s Circuital Law, a fundamental law of magnetostatics.

Statement

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

Mathematical Form

Scalar form:

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

Vector form:

dB = \[\frac{\mu_{0}}{4\pi}\frac{Id\mathbf{l}\times\mathbf{r}}{r^{3}}\]​

where

• μ0​ = permeability of free space
• I = current
• dl = current element
• r = distance from element to point
• θ = angle between dl and r

Explanation

The total magnetic field at a point due to a current-carrying conductor is obtained by integrating (summing) the contributions of all small current elements along the conductor:

B = ∫ dB

Conclusion

Thus, the Biot–Savart Law gives the magnitude and direction of the magnetic field produced by a current-carrying conductor and follows an inverse square law dependence on distance.

• 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.