5 Questions
What is the mathematical operation that describes the rate of change of a scalar field with respect to position?
Gradient
Which of Maxwell's equations relates the electric field around a closed loop to the changing magnetic flux through the loop?
Faraday's law of induction
What is the term for the circulation of a vector field around a closed curve?
Curl
Which of the following laws describes the relationship between the magnetic field and the electric current producing it in a static field?
Ampere's law
What is the term for the rate of change of a vector field with respect to position?
Del operator
Study Notes
Fields
- A scalar field is a function that assigns a numerical value to each point in space.
- A vector field is a function that assigns a vector to each point in space.
Vector Calculus Operations
- Gradient of a scalar field: a vector field that points in the direction of the maximum rate of increase of the scalar field.
- Divergence of a vector field: a scalar field that describes the rate of flux of the vector field.
- Curl of a vector field: a vector field that describes the rotation of the original vector field.
Maxwell's Equations
- Maxwell's First Equation (Gauss's Law for Electric Field): ∇ ⋅ E = ρ/ε₀, relates electric flux to charge distribution.
- Maxwell's Second Equation (Gauss's Law for Magnetic Field): ∇ ⋅ B = 0, states that magnetic charges do not exist.
- Maxwell's Third Equation (Faraday's Law of Induction): ∇ × E = -∂B/∂t, describes how changing magnetic fields induce electric fields.
- Maxwell's Fourth Equation (Ampere's Law with Maxwell's Correction): ∇ × B = μ₀J + μ₀ε₀∂E/∂t, relates magnetic fields to electric currents and displacement currents.
Electromagnetic Induction
- Electrostatics: the study of stationary electric charges and their effects.
- Magnetostatics: the study of stationary magnetic fields and their effects.
- Faraday's Law of Induction: describes how changing magnetic flux induces electric fields.
- Ampere's Law: relates magnetic fields to electric currents in static fields.
- Ampere's Law in Time-Varying Fields: includes the effect of displacement currents in time-varying fields.
Test your knowledge of electromagnetism and Maxwell's equations, including scalar and vector fields, curl, divergence, and gradient. Covering electrostatics, magnetostatics, Faraday's law, and Ampere's law in both static and time-varying fields.
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