November 6, 2024
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Moving Charges Important Formulae

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Formulae

1. **Magnetic Force on a Moving Charge**
– F = q(\mathbf{v} \times \mathbf{B})
– F: Force on the charged particle (Unit: Newton, N)
– q: Charge of the particle (Unit: Coulomb, C)
– \mathbf{v}: Velocity of the particle (Unit: meters/second, m/s)
– \mathbf{B}: Magnetic Field (Unit: Tesla, T)

2. **Magnetic Field due to a Current in a Straight Conductor**
– B = \frac{\mu_0 I}{2\pi r}
– B: Magnetic field (Unit: Tesla, T)
– \mu_0: Permeability of free space (Unit: T \cdot m/A or Tesla meter per Ampere)
– I: Current through the conductor (Unit: Ampere, A)
– r: Distance from the conductor (Unit: meter, m)

3.**Torque on a Current Loop in a Magnetic Field**
– \tau = \mathbf{m} \times \mathbf{B}
– \tau: Torque on the current loop (Unit: Newton meter, Nm)
– \mathbf{m}: Magnetic moment of the current loop (Unit: A \cdot m^2 or Ampere square meter)
– \mathbf{B}: Magnetic Field (Unit: Tesla, T)

4. **Magnetic Flux through a Surface**
– \Phi_B = \int \mathbf{B} \cdot d\mathbf{A}
– \Phi_B: Magnetic flux (Unit: Weber, Wb)
– \mathbf{B}: Magnetic Field (Unit: Tesla, T)
– d\mathbf{A}: Differential area vector (Unit: square meter, m^2)

5. **Faraday’s Law of Electromagnetic Induction**
– \mathcal{E} = -\frac{d\Phi_B}{dt}
– \mathcal{E}: Induced electromotive force (emf) (Unit: Volt, V)
– \Phi_B: Magnetic flux (Unit: Weber, Wb)
– t: Time (Unit: second, s)

6.**Biot-Savart Law**
– d\mathbf{B} = \frac{\mu_0 I \, d\mathbf{l} \times \mathbf{r}}{4\pi r^3}
– d\mathbf{B}: Differential magnetic field (Unit: Tesla, T)
– \mu_0: Permeability of free space (Unit: T \cdot m/A)
– I: Current (Unit: Ampere, A)
– d\mathbf{l}: Differential length vector of current-carrying wire (Unit: meter, m)
– \mathbf{r}: Position vector from the current element to the point in space (Unit: meter, m)

7. **Magnetic Field in a Solenoid**
– B = \mu_0 nI
– B: Magnetic field inside the solenoid (Unit: Tesla, T)
– \mu_0: Permeability of free space (Unit: T \cdot m/A)
– n: Number of turns per unit length of the solenoid (Unit: turns/meter)
– I: Current (Unit: Ampere, A)

Problems based on each concept

1. **Magnetic Force on a Charge**
– A proton is moving with a velocity of 5 \times 10^5 \, \text{m/s} perpendicular to a magnetic field of 2 \, \text{T}. Find the magnetic force on the proton.

2. **Magnetic Field due to a Current through a Straight Conductor**
– Calculate the magnetic field 10 cm from a long straight wire carrying a current of 5 A.

3. **Magnetic Field due to a Current through a Circular Loop (at the center)**
– A circular loop of radius 20 cm carries a current of 2 A. Find the magnetic field at the center of the loop.

4. **Ampère’s Circuital Law**
– A solenoid with 100 turns per cm has a radius of 2 cm and carries a current of 3 A. Use Ampère’s Law to find the magnetic field inside the solenoid.

5. **Force between Two Parallel Conductors**
– Two parallel wires 1 m long and separated by a distance of 5 cm carry currents of 10 A and 20 A respectively. Find the force between them.

6. **Biot-Savart Law**
– A short wire segment of length 0.1 m carries a current of 2 A. Calculate the magnetic field at a point 5 cm from the wire due to this segment.

7. **Torque on a Current Loop in a Magnetic Field**
– A rectangular loop with sides 10 cm and 5 cm carries a current of 3 A. It is placed in a magnetic field of 0.2 T. Find the maximum torque it can experience.

8. **Lorentz Force (Combined electric and magnetic force)**
– A charge q = 1 \times 10^{-6} \, \text{C} is moving with a velocity of 3 \times 10^5 \, \text{m/s} in an electric field of 100 N/C and a magnetic field of 0.5 T. Calculate the total force on the charge.

9. **Magnetic Field in a Solenoid**
– A solenoid of length 50 cm has 100 turns and carries a current of 4 A. Calculate the magnetic field inside the solenoid.

10. **Magnetic Flux through a Surface**
– A plane of area 0.5 m^2 is placed in a uniform magnetic field of 3 T at an angle of 60° to the field. Find the magnetic flux through the plane.

11. **Faraday’s Law of Electromagnetic Induction**
– A coil of 500 turns has a cross-sectional area of 0.1 m^2. If the magnetic field through it changes from 0 to 0.5 T in 0.2 seconds, calculate the induced emf in the coil.

12. **Lenz’s Law**
– (Conceptual) A magnet is moved towards a coil. If the north pole of the magnet is approaching the coil, will the induced current in the coil make the facing end of the coil a north pole or a south pole?

13. **Self Inductance**
– A solenoid has an inductance of 0.5 H. If the current through it is increasing at a rate of 2 A/s, what is the induced emf?

14. **Mutual Inductance between Two Coils**
– Two coils are placed close to each other. The mutual inductance between them is 0.8 H. If the current in the first coil changes from 0 to 5 A in 0.1 seconds, find the induced emf in the second coil.

Key

1. **Magnetic Force on a Charge**
– F = q(\mathbf{v} \times \mathbf{B})
F = e \times vB (since \mathbf{v} \times \mathbf{B} results in vB for perpendicular vectors)
F = (1.6 \times 10^{-19} \, \text{C})(5 \times 10^5 \, \text{m/s})(2 \, \text{T}) = 1.6 \times 10^{-13} \, \text{N}

2. **Magnetic Field due to a Current through a Straight Conductor**
– B = \frac{\mu_0 I}{2\pi r}
B = \frac{4\pi \times 10^{-7} \, \text{Tm/A} \times 5 \, \text{A}}{2\pi \times 0.1 \, \text{m}} = 1 \times 10^{-5} \, \text{T}

3. **Magnetic Field due to a Current through a Circular Loop (at the center)**
– B = \frac{\mu_0 I}{2R}
B = \frac{4\pi \times 10^{-7} \, \text{Tm/A} \times 2 \, \text{A}}{2 \times 0.2 \, \text{m}} = 2 \times 10^{-6} \, \text{T}

4. **Ampère’s Circuital Law**
– For this problem, the formula for a solenoid is more straightforward: B = \mu_0 nI
B = 4\pi \times 10^{-7} \, \text{Tm/A} \times 100 \, \text{turns/cm} \times 3 \, \text{A} = 0.0377 \, \text{T}

5. **Force between Two Parallel Conductors**
– F = \frac{\mu_0 I_1 I_2 L}{2\pi r}
F = \frac{4\pi \times 10^{-7} \, \text{Tm/A} \times 10 \, \text{A} \times 20 \, \text{A} \times 1 \, \text{m}}{2\pi \times 0.05 \, \text{m}} = 8 \times 10^{-5} \, \text{N/m}

6. **Biot-Savart Law**
– This is a complex calculation involving vector calculus. For simplicity, let’s assume we’re calculating the field at a point perpendicular to the wire segment’s midpoint. In practice, this requires integration over the wire segment.

7. **Torque on a Current Loop in a Magnetic Field**
– \tau = NIAB\sin(\theta) (assuming \theta = 90° for maximum torque)
\tau = 1 \times 3 \, \text{A} \times 0.05 \, \text{m} \times 0.1 \, \text{m} \times 0.2 \, \text{T} = 0.003 \, \text{Nm}

8. **Lorentz Force (Combined electric and magnetic force)**
– \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})
F = 1 \times 10^{-6} \, \text{C} \times (100 \, \text{N/C} + 0) = 1 \times 10^{-4} \, \text{N} (assuming v and B are perpendicular)

9. **Magnetic Field in a Solenoid**
– B = \mu_0 nI
B = 4\pi \times 10^{-7} \, \text{Tm/A} \times 200 \, \text{turns/m} \times 4 \, \text{A} = 1 \times 10^{-3} \, \text{T}

10. **Magnetic Flux through a Surface**
– \Phi_B = B \times A \times \cos(\theta)
\Phi_B = 3 \, \text{T} \times 0.5 \, \text{m}^2 \times 0.5 = 0.75 \, \text{Wb}

11. **Faraday’s Law of Electromagnetic Induction**
– \mathcal{E} = -N\frac{d\Phi_B}{dt}
Change in flux: \Delta\Phi_B = 500 \times 0.5 \, \text{T} \times 0.1 \, \text{m}^2 = 25 \, \text{Wb}
\mathcal{E} = -500 \times \frac{25 \, \text{Wb}}{0.2 \, \text{s}} = -62500 \, \text{V} (negative sign indicates opposing nature as per Lenz’s Law)

12. **Lenz’s Law**
– The induced current in the coil will oppose the motion of the north pole, so it will make the facing end of the coil a **North pole**.

13. **Self Inductance**
– \mathcal{E} = -L \frac{dI}{dt}
\mathcal{E} = -0.5 \, \text{H} \times 2 \, \text{A/s} = -1 \, \text{V}

14. **Mutual Inductance between Two Coils**
– \mathcal{E}_2 = -M \frac{dI_1}{dt}
\mathcal{E}_2 = -0.8 \, \text{H} \times \frac{5 \, \text{A}}{0.1 \, \text{s}} = -40 \, \text{V}

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