### Shortcut Methods

**JEE Main Level:**

**Magnetic field due to a long straight wire:**

$$B = \frac{\mu_0 I}{2\pi d}$$

Where:

- B is the magnetic field (in Tesla)
- I is the current (in Ampere)
- d is the distance from the wire (in meters)
- µ0 is the permeability of vacuum (( 4\pi \times 10^{-7}) H/m)

**Magnetic field at the center of a circular loop:**

$$B = \frac{\mu_0 I}{2R}$$

Where:

- B is the magnetic field (in Tesla)
- I is the current (in Ampere)
- R is the radius of the loop (in meters)

**Magnetic field inside a solenoid:**

$$B = \mu_0 nI$$

Where:

- B is the magnetic field (in Tesla)
- n is the number of turns per unit length (in turns/meter)
- I is the current (in Ampere)

**Speed of a charged particle in a magnetic field:**

$$v = \frac{Br}{m}$$

Where:

- v is the speed of the particle (in meters per second)
- B is the magnetic field (in Tesla)
- r is the radius of the circular path (in meters)
- m is the mass of the particle (in kilograms)

**CBSE Board Level:**

**Magnetic field due to a long straight wire:**

$$B = \frac{\mu_0 I}{2\pi d}$$

Where:

- B is the magnetic field (in Tesla)
- I is the current (in Ampere)
- d is the distance from the wire (in meters)
- µ0 is the permeability of vacuum (( 4\pi \times 10^{-7}) H/m)

**Magnetic field at the center of a circular loop:**

$$B = \frac{\mu_0 I}{2R}$$

Where:

- B is the magnetic field (in Tesla)
- I is the current (in Ampere)
- R is the radius of the loop (in meters)

**Magnetic field inside a solenoid:**

$$B = \mu_0 nI$$

Where:

- B is the magnetic field (in Tesla)
- n is the number of turns per unit length (in turns/meter)
- I is the current (in Ampere)

**Speed of a charged particle in a magnetic field:**

$$v = \frac{Br}{m}$$

Where:

- v is the speed of the particle (in meters per second)
- B is the magnetic field (in Tesla)
- r is the radius of the circular path (in meters)
- m is the mass of the particle (in kilograms)

**Additional Numerical:**

**Magnetic field due to a long straight wire:**

$$B = \frac{\mu_0 I}{2\pi d}$$

Where:

- B is the magnetic field (in Tesla)
- I is the current (in Ampere)
- d is the distance from the wire (in meters)
- µ0 is the permeability of vacuum = (( 4\pi \times 10^{-7}) H/m)

**Magnetic field at the center of a square loop:**

$$B = \frac{\mu_0 I}{2R}$$

Where:

- B is the magnetic field (in Tesla)
- I is the current (in Ampere)
- R is the radius of the loop (in meters)
- In case of square loop, (R = \frac{a}{\sqrt{2}}), where ‘a’ is the side length of the square.

**Magnetic field inside a toroid:**

$$B = \mu_0 nI$$

Where:

- B is the magnetic field (in Tesla)
- n is the number of turns per unit length (in turns/meter)
- I is the current (in Ampere)

**Mass of a charged particle in a magnetic field:**

$$m = \frac{Bqr}{v}$$

Where:

- m is the mass of the particle (in kilograms)
- B is the magnetic field (in Tesla)
- q is the charge of the particle (in Coulombs)
- r is the radius of the circular path (in meters)
- v is the speed of the particle (in meters per second)