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Basic Electromagnetic Equations

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Faraday's Law of Induction

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Equation:

E=dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}
; Variables: E\mathcal{E} (induced electromotive force), ΦB\Phi_B (magnetic flux); Represents how a changing magnetic field can induce an electromotive force in a coil.

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Gauss's Law for Magnetism

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Equation:

BdA=0\oint \vec{B} \cdot d\vec{A} = 0
; Variables: B (magnetic field), dAd\vec{A} (infinitesimal area element); Represents the net magnetic flux through a closed surface is always zero indicating no magnetic monopoles.

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Ampère's Circuital Law

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Equation:

Bdl=μ0Ienc\oint \vec{B} \cdot d\vec{l} = \mu_0I_{enc}
; Variables: B (magnetic field), dld\vec{l} (infinitesimal length of wire), μ0\mu_0 (permeability of free space), IencI_{enc} (enclosed current); Represents the magnetic field around a current-carrying conductor.

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Gauss's Law for Electricity

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Equation:

EdA=Qencϵ0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}
; Variables: E (electric field), dAd\vec{A} (infinitesimal area element), QencQ_{enc} (enclosed charge), ϵ0\epsilon_0 (vacuum permittivity); Represents the electric flux through a closed surface is proportional to the charge enclosed.

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Coulomb's Law

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Equation:

F=keq1q2r2F = k_e \frac{q_1 q_2}{r^2}
; Variables: F (force), kek_e (Coulomb's constant), q1q_1 and q2q_2 (charges), r (distance); Represents the electrostatic force between two charged particles.

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Maxwell's Equations

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Equation: Description of four fundamental equations; Variables: Varies per equation (includes electric field E, magnetic field B, etc.); Represents a complete set of laws of classical electromagnetism.

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Biot-Savart Law

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Equation:

B=μ04πIdl×r^r2\vec{B} = \frac{\mu_0}{4\pi} \int \frac{Id\vec{l} \times \hat{r}}{|r|^2}
; Variables: B (magnetic field), μ0\mu_0 (permeability of free space), I (current), dld\vec{l} (infinitesimal element of current carrying wire), r^\hat{r} (unit vector from wire element to point P), r (distance); Represents the magnetic field created at a point due to a small current element.

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The Continuity Equation

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Equation:

ρt+(ρv)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0
; Variables: ρ\rho (charge density), t (time), v\vec{v} (velocity of the charge flow); Represents the principle of charge conservation in a volume.

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Ohm's Law

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Equation: V=IRV = IR; Variables: V (voltage), I (current), R (resistance); Represents how the voltage in a circuit is the product of the current and the resistance.

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Lorentz Force

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Equation:

F=q(E+v×B)\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})
; Variables: F (force), q (charge), E (electric field), v (velocity), B (magnetic field); Represents the force experienced by a charged particle moving in electric and magnetic fields.

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