Maxwell's Equations and Displacement Current
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Questions and Answers

What is the term used to describe the additional term in Ampere's law that ensures divergence of current density is zero?

  • Magnetic current
  • Displacement current (correct)
  • Electric current
  • Conduction current

    • Which term in Ampere's law represents the effect of a varying electric field causing a magnetic field even without an electric current?

  • J
  • Jd (correct)
  • Conduction current
  • Displacement current

    • According to the Helmholtz theorem, what conditions must be satisfied to determine an electromagnetic field at any point?

  • The field must have a varying magnetic flux.
  • The field must have zero divergence and curl.
  • The field must vanish far from its source. (correct)
  • The field must be constant throughout space.

    • In vacuum, what is the value of the divergence of the electric field according to Maxwell's equations?

    <p>0</p> Signup and view all the answers

    What happens to Maxell's equations in vacuum if there is no free charge or current present?

    <p>They simplify to a subset of equations.</p> Signup and view all the answers

    Which term in Ampere's law accounts for the magnetostatic effects of currents?

    <p>( \mu_0 J )</p> Signup and view all the answers

    Which parameter is directly proportional to the current density in a conductor?

    <p>Electric field</p> Signup and view all the answers

    What is the characteristic time (τ) defined as in a conducting medium?

    <p>Time after which charge density reduces to 1/e</p> Signup and view all the answers

    What is the relationship between current (I), voltage (V), and resistance (R) in a conducting medium?

    <p>$I= \frac{V \cdot A}{L}$</p> Signup and view all the answers

    Which equation represents the charge conservation principle in a conducting medium?

    <p>$\nabla \cdot J = -\nabla(\sigma E)$</p> Signup and view all the answers

    What quantity does the wave impedance in free space (Z0) represent?

    <p>Impedance of electromagnetic waves in vacuum</p> Signup and view all the answers

    What does Ohm's law state about the relationship between current density and electric field in a conductor?

    <p>$J$ is proportional to $E$</p> Signup and view all the answers

    What does a complex k vector in a conducting medium indicate?

    <p>A phase lag between E and B</p> Signup and view all the answers

    What is the consequence of a complex wave vector in a conducting medium?

    <p>E and B vectors are out of phase</p> Signup and view all the answers

    How is the magnitude of a complex k vector represented?

    <p>|k| = k+ + k- + i(2k+ k-)</p> Signup and view all the answers

    For good conductors, what is the relationship between the two components of the complex wave vector (k+ and k-)?

    <p>k+ = ω^2 σ / μσω = k-</p> Signup and view all the answers

    What does Maxwell's third equation indicate for E and B in a conducting medium?

    <p>E lags B by 90 degrees</p> Signup and view all the answers

    In the context of poor conductors, what is the relationship between σ, ω, μ, and ε that affects the wave vector components?

    <p>|σ| &lt;&lt; |ωε|</p> Signup and view all the answers

    What is a fundamental limitation of Ampere's law?

    <p>It is not valid for steady current</p> Signup and view all the answers

    How did Maxwell resolve the discrepancy in Ampere's law?

    <p>By adding a term for electric field in Ampere's law</p> Signup and view all the answers

    What is the main improvement in Ampere-Maxwell's law compared to Ampere's law?

    <p>Consideration of time-varying electric fields</p> Signup and view all the answers

    Which equation represents the continuity equation in electrodynamics?

    <p>$\nabla \cdot J = -\nabla \cdot \frac{\partial E}{\partial t}$</p> Signup and view all the answers

    Why does Ampere's circuital law show inconsistency when a capacitor is charged?

    <p>As a result of displacement current not being considered</p> Signup and view all the answers

    What did Maxwell introduce to explain the displacement current?

    <p>Rate of change of electric field</p> Signup and view all the answers

    What is the operator ∇ called?

    <p>Del vector operator</p> Signup and view all the answers

    In terms of dot product and cross product, the ∇ operator behaves like a:

    <p>Vector</p> Signup and view all the answers

    What does the divergence of a vector field at a point represent?

    <p>Flux generation per unit volume</p> Signup and view all the answers

    What does ∬A⋅dS represent in the context of a vector field A at a closed surface S?

    <p>Flux through the surface</p> Signup and view all the answers

    What does the curl of a vector A represent?

    <p>Maximum circulation per unit area</p> Signup and view all the answers

    For Cartesian coordinates, what is the expression for ∇×A?

    <p>$iˆ \frac{\partial}{\partial x} Ax + ˆj \frac{\partial}{\partial y} Ay + kˆ \frac{\partial}{\partial z} Az$</p> Signup and view all the answers

    Study Notes

    Vector Operators

    • The operator ∇ is called "del" and is not a vector, but behaves like a vector in terms of dot product and cross product.

    Divergence of a Vector

    • The divergence of a vector field at a point represents the flux generation per unit volume centered at that point.
    • The divergence of a vector field is defined as: ∇⋅A = (∂Ax/∂x) + (∂Ay/∂y) + (∂Az/∂z)
    • The divergence of a vector field can be positive, negative, or zero, indicating source, sink, or no flux at a point.

    Curl of a Vector

    • The curl of a vector is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area.
    • The curl of a vector is defined as: ∇×A = (∂Az/∂y) - (∂Ay/∂z) i + (∂Ax/∂z) - (∂Az/∂x) j + (∂Ay/∂x) - (∂Ax/∂y) k
    • The curl of a vector represents the rotation of the vector field.

    Ampere's Law

    • Ampere's law states that the circulation of the magnetic field around a closed loop is proportional to the current flowing through the loop.
    • The law is limited to magnetostatics and only applies to steady currents.
    • Maxwell's correction to Ampere's law introduces the concept of displacement current, which allows for time-varying electric fields.

    Maxwell's Equations

    • Maxwell's equations are a set of four equations that describe the behavior of electric and magnetic fields.
    • The equations are:
      • Gauss's law: ∇⋅E = ρ/ε₀
      • Gauss's law for magnetism: ∇⋅B = 0
      • Faraday's law of induction: ∇×E = -∂B/∂t
      • Ampere's law with Maxwell's correction: ∇×B = μ₀J + μ₀ε₀∂E/∂t
    • These equations form the basis of classical electromagnetism.

    Maxwell's Equations in Vacuum

    • In vacuum, the Maxwell's equations take the form:
      • ∇⋅E = 0
      • ∇⋅B = 0
      • ∇×E = -∂B/∂t
      • ∇×B = μ₀ε₀∂E/∂t
    • The solutions to these equations describe the electric and magnetic fields in free space.

    Solution of Maxwell's Equations in Vacuum

    • The solution to the Maxwell's equations in vacuum involves the electric and magnetic fields, E and B, which are perpendicular to each other and to the direction of propagation.
    • The wave impedance in free space is Z₀ = √(μ₀/ε₀) ≈ 377 Ω.

    Maxwell's Equations in a Conducting Medium

    • In a conducting medium, the Maxwell's equations take the form:
      • ∇⋅E = ρ/ε
      • ∇⋅B = 0
      • ∇×E = -∂B/∂t
      • ∇×B = μJ + με∂E/∂t
    • The solutions to these equations describe the electric and magnetic fields in a conducting medium.

    Solution of Maxwell's Equations in a Conducting Medium

    • The solution to the Maxwell's equations in a conducting medium involves the electric and magnetic fields, E and B, which are not in phase.
    • The wave vector k is a complex quantity, indicating a phase lag between E and B.
    • The magnitude of the complex wave vector can be written as k = k+ + ik-, where k+ and k- are real and imaginary parts of the wave vector.

    Consequence of Complex Wave Vector in Conducting Medium

    • The complex wave vector indicates a phase lag between the electric and magnetic fields in a conducting medium.
    • For good conductors, the phase lag is significant, while for poor conductors, the phase lag is negligible.
    • The complex wave vector has important implications for the behavior of electromagnetic waves in conducting media.

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    Description

    Learn about Maxwell's equations and the concept of displacement current. Understand how the modification in Ampere's law accounts for the relationship between changing electric fields and magnetic fields.

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