Magnetic Effects of Current & Magnetism Cheatsheet

Key Concepts & Formulas

• Biot-Savart Law: \( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2} \)
• Magnetic Field of a Straight Wire: \( B = \frac{\mu_0 I}{2\pi r} \)
• Magnetic Field at Center of a Loop: \( B = \frac{\mu_0 I}{2R} \)
• Ampere's Circuital Law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \)
• Lorentz Force: \( \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \)
• Force on a Current-Carrying Wire: \( \vec{F} = I(\vec{l} \times \vec{B}) \)
• Magnetic Dipole Moment: \( \vec{\mu} = I\vec{A} \)
• Torque on a Dipole: \( \vec{\tau} = \vec{\mu} \times \vec{B} \)
• Magnetic Flux: \( \Phi_B = \int \vec{B} \cdot d\vec{A} \)
• Solenoid Field: \( B = \mu_0 n I \)

Quick Reference Table

Concept	Formula	Explanation
Biot-Savart Law	\( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2} \)	Finds magnetic field from a current element.
Force on Charge	\( \vec{F} = q(\vec{v} \times \vec{B}) \)	Force on a moving charge in a magnetic field.
Ampere's Law	\( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \)	Relates magnetic field around a closed loop to the current passing through it.
Torque on Loop	\( \vec{\tau} = \vec{\mu} \times \vec{B} \)	Torque experienced by a current loop in a magnetic field.
Cyclotron Frequency	\( \omega = \frac{qB}{m} \)	Angular frequency of a charged particle's circular motion.
Magnetic Field of Toroid	\( B = \frac{\mu_0 N I}{2\pi r} \)	Magnetic field inside a toroid.

Tips for Solving Problems

Magnetic Effects Quick Tips

• Right-Hand Rule: Point your thumb in the direction of current (I), and your fingers curl in the direction of the magnetic field (B).
• Force on a Wire: For \( \vec{F} = I(\vec{l} \times \vec{B}) \), use the right-hand palm rule to find force direction.
• Solenoid vs. Toroid: Solenoid's field is uniform and parallel to its axis, while a toroid's field is contained within its core.
• Cyclotron Motion: A charged particle moves in a circular path in a uniform magnetic field.
• Magnetic Flux: The amount of magnetic field lines passing through a given area. Maximum when B is parallel to the area vector.

Magnetic Effects Speed Quiz

Test your speed with 5 magnetic effects questions! You have 30 seconds per question.