Moving Charges and Magnetism – CBSE NCERT Study Resources

The Biot-Savart Law describes the magnetic field generated by a steady current. It states that the magnetic field dB due to a current element Idl is directly proportional to the current and inversely proportional to the square of the distance.

• For a straight conductor, integrating the law gives B = (μ₀I)/(4πa) (sinθ₁ + sinθ₂), where a is the perpendicular distance.
• Ampere’s Law simplifies calculations for symmetric systems like solenoids but lacks precision for arbitrary shapes.

While Biot-Savart is versatile, Ampere’s Law is computationally efficient for symmetrical cases. Both are experimentally validated.

A cyclotron uses a magnetic field to bend charged particles into circular paths and an electric field to accelerate them. The frequency remains constant due to the non-relativistic assumption.

• [Diagram: Two D-shaped dees with alternating voltage and perpendicular magnetic field]
• Electrons gain relativistic mass, causing desynchronization with the electric field.

Cyclotrons are ideal for protons/heavy ions but fail for electrons, which require synchrotrons. Our textbook shows its medical use in radiotherapy.

The force per unit length between parallel conductors is F/L = (μ₀I₁I₂)/(2πd), where d is separation. Attraction/repulsion depends on current direction.

• When I₁ = I₂ = 1A and d = 1m, F/L = 2×10⁻⁷ N/m, defining the ampere.
• Example: Currents in power lines exhibit this force.

This principle underpins the SI definition of current. It’s experimentally verified using current balances.

A charged particle in a magnetic field follows a circular path (F = qvB). With an electric field, the trajectory becomes helical or linear, depending on alignment.

• In a pure B-field, radius r = mv/qB.
• Crossed fields (E⊥B) create a straight path when v = E/B (Hall effect).

This explains devices like mass spectrometers. Our textbook shows applications in CRT displays.

Torque (τ = NIABsinθ) acts on a current loop in a B-field, where N is turns, A is area, and θ is the angle between normal and field.

• In galvanometers, a spring counterbalances torque, giving θ ∝ I.
• Example: Moving coil meters use this for current measurement.

This principle enables sensitive current detection. Modern digital meters still rely on its foundational physics.

The Biot-Savart Law states that the magnetic field dB due to a current element Idl is directly proportional to the current and the sine of the angle between dl and the position vector r. We studied its mathematical form: dB = (μ₀/4π) (Idl × r̂)/r².

For a circular loop of radius R, integrating the Biot-Savart Law gives the field at the center as B = (μ₀I)/2R. Our textbook shows this by considering symmetry, where all dB components add up along the axis.

This derivation assumes a thin wire and steady current. Real-world deviations occur if the loop isn't perfectly circular or if the current varies.

A cyclotron accelerates charged particles using a high-frequency alternating electric field and a perpendicular magnetic field. We studied that particles spiral outward due to the Lorentz force, gaining energy with each half-cycle.

The relativistic limit arises when particle mass increases with velocity, desynchronizing the electric field. Modern cyclotrons use variable frequency (synchrocyclotrons) or adjust the magnetic field (isochronous cyclotrons) to compensate.

While effective for protons, cyclotrons become impractical for electrons due to rapid relativistic effects, leading to the development of linear accelerators.

Torque (τ) on a current loop is given by τ = NIABsinθ, where θ is the angle between the magnetic moment (μ) and field B.

• (a) Parallel: θ = 90°, so τ = NIAB (maximum torque)
• (b) Perpendicular: θ = 0°, so τ = 0 (stable equilibrium)

This explains why galvanometer coils rotate to align with the field. Our textbook shows practical applications in electric motors where commutators maintain θ ≈ 90° for continuous rotation.

Ampere's Law states that the line integral of B around a closed loop equals μ₀I_enc. We studied its use for highly symmetric systems where B is constant along the path.

• Infinite straight wire: Choosing a circular Amperian loop gives B = (μ₀I)/2πr
• Solenoid: Applying the law to a rectangular loop yields B = μ₀nI inside

While powerful, the law fails for asymmetric distributions like a finite wire segment, requiring Biot-Savart calculations instead.

The force arises due to the interaction between the magnetic field (B₁ = μ₀I₁/2πd) of conductor 1 and current I₂ in conductor 2.

Using F = I₂lB₁, we get force per unit length as F/l = (μ₀I₁I₂)/2πd. Our textbook shows this leads to attraction for parallel currents and repulsion for antiparallel currents.

This principle defines the ampere in SI units: 1A is the current producing 2×10^-7 N/m force at 1m separation. This bridges electromagnetism and mechanical measurements.

A cyclotron is a device used to accelerate charged particles, such as protons or electrons, to high energies using a combination of electric and magnetic fields. It consists of two hollow semicircular electrodes called dees placed in a uniform magnetic field perpendicular to their plane. The charged particles are injected near the center and accelerated by an alternating electric field between the dees.

Working Principle:
1. A charged particle is released at the center of the dees.
2. A magnetic field causes the particle to move in a circular path.
3. An alternating electric field accelerates the particle each time it crosses the gap between the dees.
4. The radius of the path increases as the particle gains energy, leading to a spiral trajectory.

Derivation of Cyclotron Frequency:
The centripetal force is provided by the magnetic force:
$$ qvB = \frac{mv^2}{r} $$
Rearranging gives the angular frequency (ω):
$$ ω = \frac{v}{r} = \frac{qB}{m} $$
The cyclotron frequency (f) is:
$$ f = \frac{ω}{2π} = \frac{qB}{2πm} $$

Limitations:

• Relativistic effects: As particles approach the speed of light, their mass increases, causing a mismatch with the alternating field frequency.
• Size and cost: High-energy cyclotrons require large magnets and are expensive to build.
• Limited to charged particles: Neutral particles cannot be accelerated.

The Biot-Savart law states that the magnetic field dB due to a current element Idl at a point is:
dB = (μ₀/4π) (Idl × r)/r³
where μ₀ is permeability of free space, r is the position vector from the element to the point, and × denotes the cross product.

Derivation for Circular Loop:
1. Consider a circular loop of radius R carrying current I.
2. The magnetic field at the center is the sum of contributions from all dl elements.
3. For each element, dl and r are perpendicular, so |dl × r| = dl·R.
4. Integrating over the loop:
B = (μ₀I/4πR²) ∮dl = (μ₀I/4πR²) (2πR)
Thus, B = μ₀I/2R at the center.

Variation Along Axis:
The field at a distance x from the center along the axis is:
Bₓ = μ₀IR²/2(R² + x²)^(3/2)
This shows:
1. The field is maximum at the center (x = 0).
2. It decreases as we move away from the center along the axis.
3. At large distances (x >> R), Bₓ ≈ μ₀IR²/2x³ (dipole-like behavior).

A moving coil galvanometer is a device used to detect and measure small electric currents based on the principle that a current-carrying coil placed in a magnetic field experiences a torque.

Construction:
1. It consists of a rectangular coil suspended between the poles of a permanent magnet with a cylindrical soft iron core.
2. The coil is wound on a lightweight aluminum frame and attached to a phosphor-bronze strip.
3. A mirror is fixed to the suspension to measure deflection using a lamp and scale arrangement.

Working:
When current I flows through the coil, it experiences a torque τ = NIAB (N=number of turns, A=area, B=magnetic field). This torque is balanced by the restoring torque kθ (k=restoring couple per unit twist, θ=deflection angle). At equilibrium:
NIAB = kθ
Thus, θ = (NAB/k)I, showing deflection is proportional to current.

Current Sensitivity (Sᵢ):
Defined as θ/I = NAB/k
To increase Sᵢ:
1. Increase N, A, or B: More turns, larger coil area, or stronger magnet.
2. Decrease k: Use a weaker suspension fiber (reduces restoring torque).
However, increasing sensitivity may reduce the instrument's durability and speed of response.

The Biot-Savart Law states that the magnetic field (dB) due to a small current element (Idl) at a point is:
$$ dB = \frac{μ₀}{4π} \frac{Idl sinθ}{r^2} $$
where μ₀ is permeability of free space, θ is the angle between Idl and the position vector r.

Derivation for Circular Loop:
1. Consider a circular loop of radius R carrying current I.
2. For each element Idl, the angle θ is 90°, so sinθ = 1.
3. The magnetic field at the center due to one element is:
$$ dB = \frac{μ₀}{4π} \frac{Idl}{R^2} $$
4. Integrating over the entire loop:
$$ B = \frac{μ₀}{4π} \frac{I}{R^2} ∫dl = \frac{μ₀}{4π} \frac{I}{R^2} (2πR) $$
5. Simplifying gives:
$$ B = \frac{μ₀I}{2R} $$

Application in Helmholtz Coils:
Helmholtz coils consist of two identical circular coils placed coaxially at a distance equal to their radius. They produce a nearly uniform magnetic field in the central region, which is useful in experiments like electron deflection and magnetic field calibration. The combined field is:
$$ B_{total} = \frac{8μ₀NI}{5^{3/2}R} $$
where N is the number of turns in each coil.

A cyclotron is a device used to accelerate charged particles, such as protons or deuterons, to high energies using a combination of electric and magnetic fields. It operates on the principle that a charged particle moving perpendicular to a uniform magnetic field experiences a Lorentz force, causing it to move in a circular path. The particle gains energy each time it passes through an alternating electric field between two hollow dees (semi-circular electrodes).

Construction: A cyclotron consists of:

• Two dees (D1 and D2) placed in a vacuum chamber.
• A strong electromagnet to provide a uniform magnetic field perpendicular to the plane of the dees.
• A high-frequency alternating voltage source connected across the dees.
• An ion source at the center to inject charged particles.

Working:

• The charged particle is released at the center and accelerated by the electric field between the dees.
• The magnetic field forces the particle into a circular path.
• Each time the particle crosses the gap between the dees, the polarity of the voltage reverses, providing acceleration.
• The radius of the path increases as the particle gains energy, forming a spiral.

Limitations: Cyclotrons cannot accelerate electrons (due to relativistic mass increase) or neutral particles. They are also limited by the size of the magnet and frequency constraints.

Applications: Cyclotrons are used in nuclear physics research, medical isotope production (e.g., for PET scans), and cancer treatment (proton therapy).

The force per unit length between two parallel current-carrying conductors can be derived using Ampère's law and the concept of magnetic fields due to currents.

Derivation:

Definition of Ampere: The ampere is defined as the constant current which, if maintained in two straight parallel conductors of infinite length and negligible cross-section, placed 1 meter apart in a vacuum, would produce a force of 2 × 10⁻⁷ newtons per meter of length. This directly follows from the derived expression.

A cyclotron is a device used to accelerate charged particles (like protons, deuterons) to high energies using a combination of electric and magnetic fields. It operates on the principle that a charged particle moving perpendicular to a uniform magnetic field follows a circular path, and an alternating electric field can accelerate it repeatedly.

Construction:
1. Two hollow semicircular metal chambers called dees (D1 and D2) are placed in a vacuum chamber.
2. The dees are connected to a high-frequency alternating voltage source.
3. A strong uniform magnetic field is applied perpendicular to the plane of the dees using electromagnets.
4. A source of charged particles (e.g., protons) is placed at the center.

Working:
1. The charged particle is injected at the center and accelerated by the electric field between the dees.
2. Inside the dees, the magnetic field causes the particle to move in a semicircular path.
3. The alternating voltage is synchronized such that the particle is accelerated each time it crosses the gap.
4. The radius of the path increases with energy, and the particle spirals outward until it exits.

Cyclotron Frequency:
The time taken for a half-circle is t = πr/v, where r = mv/qB (from centripetal force balance).
Thus, the frequency f = 1/T = qB/2πm, which is independent of velocity and radius.

Limitations:
1. Relativistic effects: At high speeds, mass increases, disrupting synchronization.
2. Electron acceleration is impractical due to their small mass and high energy loss.
3. Requires a uniform magnetic field, which is challenging for large sizes.

A cyclotron is a device used to accelerate charged particles, such as protons or deuterons, to high energies using a combination of electric and magnetic fields. It operates on the principle that a charged particle moving perpendicular to a uniform magnetic field follows a circular path, and an alternating electric field is used to accelerate the particle in steps.

Working Principle:

• The cyclotron consists of two hollow semicircular electrodes called dees (D1 and D2), placed in a vacuum chamber.
• A strong uniform magnetic field acts perpendicular to the plane of the dees.
• An alternating high-frequency electric field is applied between the dees.
• The charged particle is injected near the center and accelerated each time it crosses the gap between the dees.
• The magnetic field keeps the particle in a spiral path of increasing radius.

Derivation of Cyclotron Frequency:
The centripetal force required for circular motion is provided by the magnetic force:
qvB = mv²/r
Simplifying, the angular velocity (ω) is:
ω = v/r = qB/m
The cyclotron frequency (f) is:
f = ω/2π = qB/2πm

Limitations:

• At high velocities, relativistic effects increase the mass of the particle, causing a mismatch with the alternating electric field frequency.
• Maintaining a uniform magnetic field over a large area is challenging.
• It cannot accelerate neutral particles or electrons efficiently due to energy loss via radiation.

Diagram: (A labeled diagram of a cyclotron showing the dees, magnetic field direction, particle path, and alternating voltage source should be included.)

Biot-Savart's law gives the magnetic field (dB) produced by a small current element (Idl) at a point. Mathematically:
dB = (μ₀/4π) (Idl × r̂)/r²
where μ₀ is permeability of free space, r̂ is the unit vector from the current element to the point, and r is the distance.

Derivation for a Circular Loop:
Consider a circular loop of radius R carrying current I. For a small element dl, the magnetic field at the center is:
dB = (μ₀/4π) (Idl sinθ)/R²
Since θ = 90° (angle between dl and r̂), sinθ = 1.
Integrating over the entire loop:
B = ∫dB = (μ₀I/4πR²) ∫dl
The total length of the loop is 2πR, so:
B = (μ₀I/4πR²) × 2πR = μ₀I/2R

Direction of Magnetic Field:
The direction is given by the right-hand thumb rule. If the fingers curl in the direction of current, the thumb points in the direction of the magnetic field at the center. For a circular loop, the field is perpendicular to the plane of the loop.

Additional Insight:
The field at the center of a circular loop is stronger than that of a straight wire because the circular geometry concentrates the field lines at the center.

A cyclotron is a device used to accelerate charged particles to high energies using a combination of electric and magnetic fields. It operates on the principle that a charged particle moving perpendicular to a magnetic field experiences a Lorentz force, causing it to move in a circular path. The electric field is used to increase the particle's energy in steps.

Construction: A cyclotron consists of two hollow semicircular metal chambers called dees (D1 and D2) placed in a strong uniform magnetic field perpendicular to their plane. The dees are connected to a high-frequency alternating voltage source. The entire setup is enclosed in a vacuum chamber to prevent collisions with air molecules.

Working:
1. A charged particle (e.g., proton) is injected near the center of the dees.
2. The magnetic field causes the particle to move in a circular path.
3. Each time the particle crosses the gap between the dees, the alternating electric field accelerates it.
4. The radius of the path increases as the particle gains energy, forming a spiral trajectory.
5. The particle is ejected when it reaches the outer edge with high kinetic energy.

Limitations:
- Cannot accelerate electrons (due to relativistic mass increase).
- Requires very high magnetic fields for heavy particles.
- Energy is limited by the size of the dees and magnetic field strength.

Applications:
- Used in nuclear physics experiments.
- Production of radioisotopes for medical imaging (e.g., PET scans).
- Cancer treatment through particle therapy.

The force between two parallel current-carrying conductors can be derived using Ampère's force law and the concept of magnetic fields due to currents.

Derivation:
1. Consider two infinitely long parallel conductors separated by distance d, carrying currents I₁ and I₂.
2. The magnetic field (B₁) due to I₁ at the location of the second conductor is given by: B₁ = (μ₀I₁)/(2πd).
3. The force experienced by the second conductor (F₂) is: F₂ = I₂L × B₁ (where L is length).
4. Since the field and current are perpendicular, the magnitude is: F₂ = I₂LB₁sin90° = I₂L(μ₀I₁)/(2πd).
5. Thus, force per unit length: F/L = (μ₀I₁I₂)/(2πd).

Definition of Ampere:
The ampere is defined based on this expression. When I₁ = I₂ = 1A and d = 1m, the force per unit length is exactly 2 × 10^-7 N/m. This precise measurement forms the basis for the SI definition of the ampere.

Significance:
- Shows that currents in the same direction attract, while opposite currents repel.
- Fundamental for understanding electromagnetic interactions in devices like motors and transformers.