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.

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

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.

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 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.

A toroid is a hollow circular ring (like an anchor ring) on which a large number of turns of insulated wire are closely wound. It is essentially a straight solenoid bent into a closed circular shape, forming an "endless solenoid."

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.

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

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

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.

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

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

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 \[\vec B\]

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

m = IA (or m = NIA)

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

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

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

τ = NIAB

Deflection Relation:

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

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

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

F_m​ = q(v × B)

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

F = I ∫ dl × 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

Magnetic Field Inside a Long Solenoid:

B = μ_0​ni

Magnetic Field Outside an Ideal Solenoid:

B = 0

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

F = I ∮ dl × 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.

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.

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)`

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.

• A cyclotron uses a small alternating electric field for repeated acceleration and a constant magnetic field to bring the particle back for the next push
• The time for one semicircular pass is independent of speed — this is the entire secret behind why the fixed-frequency voltage keeps working as the particle speeds up
• Resonance condition: applied frequency = qB / 2πm
• Maximum kinetic energy depends on the square of the dee radius and the square of the magnetic field: K = q²B²R² / 2m
• Cyclotrons cannot accelerate electrons or neutral particles
• Real machines correct for relativistic effects using synchro-cyclotrons (frequency decreases as mass increases)

• 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).

• A circular current loop produces a magnetic field whose axial value is B = \[\frac{\mu_0IR^2}{2(x^2+R^2)^{3/2}}\].
• At the centre of the loop (x = 0), this simplifies to B₀ = \[\frac {μ_0I}{2R}\]​, and for N turns, B₀ = \[\frac {μ_0NI}{2R}\].
• Perpendicular field components from opposite points on the loop cancel; only axial components add up.
• Direction follows the right-hand thumb rule; one face of the loop acts as a north pole, the other as a south pole.
• Straight wire segments (as in a semicircular arc problem) contribute zero field at a point lying on the line of the wire itself.

• 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.