Shortcuts and Tricks for Solving Numerical Problems

Integers

• Solenoids: Use the formula $$B=\mu_0nI/l$$, where B is the magnetic field strength, $$\mu_0$$ is the permeability of free space $$(4\times10^{-7}\text{ T}\cdot\text{m/A})$$, n is the number of turns in the solenoid, I is the current flowing through the solenoid, and l is the length of the solenoid.

• Toroids: Use the formula $$B=\mu_0nI/(2\pi r)$$, where B is the magnetic field strength, $$\mu_0$$ is the permeability of free space, n is the number of turns in the toroid, I is the current flowing through the toroid, and r is the radius of the toroid.

• Magnetic force between two long straight wires: Use the formula $$F=\frac{\mu_0I_1I_2}{2\pi d}$$, where F is the magnetic force between the wires, $$\mu_0$$ is the permeability of free space, I1 is the current flowing through the first wire, I2 is the current flowing through the second wire, and d is the distance between the wires.

Fractions

• Magnetic field strength inside a long straight wire: Use the formula $$B=\frac{\mu_0I}{2\pi d}$$, where B is the magnetic field strength, $$\mu_0$$ is the permeability of free space, I is the current flowing through the wire, and d is the distance from the wire.

Decimals

• Magnetic field strength of a circular loop: Use the formula $$B=\frac{\mu_0I}{4\pi r}\left[\sin(\theta_1)+\sin(\theta_2)\right]$$, where B is the magnetic field strength, $$\mu_0$$ is the permeability of free space, I is the current flowing through the loop, r is the radius of the loop, and $$\theta_1 \ \text{and} \ \theta_2$$ are the angles between the line from the center of the loop to the observation point and the normal to the loop at that point.

Powers

• Magnetic moment of a current loop: Use the formula $$\mu=IA$$, where $$\mu$$ is the magnetic moment, I is the current flowing through the loop, A is the area of the loop.

Algebraic expressions

• Magnetic field strength of a solenoid: Use the formula $$B=nI\mu_0$$, where B is the magnetic field strength inside the solenoid, n is the number of turns in the solenoid per unit length, I is the current flowing through the solenoid, and $$\mu_0$$ is the permeability of free space.

Equations

• Ampere’s law: (\sum Bdl=\mu_0I), where $$\mu_0$$ is the permeability of free space, I is the current passing through the loop, and the summation (\sum\space Bdl) is around a closed loop.

• Magnetic field strength of a toroid: Use the formula $$B=nI\frac{\mu_0}{2\pi r}$$, where B is the magnetic field strength inside the toroid, n is the number of turns in the toroid, I is the current flowing through the toroid, r is the radius of the toroid, and $$\mu_0$$ is the permeability of free space.

Graphs

• Magnetic field strength vs. distance from a long straight wire: The graph shows that the magnetic field strength decreases with increasing distance from the wire.

• Magnetic field strength vs. angle from a circular loop: The graph shows that the magnetic field strength is maximum at the center of the loop and decreases with increasing angle from the center.

Tables

• Comparison of different types of magnetic fields: The table compares the different types of magnetic fields, their equations, and their applications.