3. (8 points) We can get some extra practice with index notation by exploring Maxwell's equations, which govern electromagnetics. (It's worth being familiar with these equations in case you work in radar or communications some day.) These equations are: ? · E = ?/?? (5) ? · B = 0 (6) ? × E = -?B/?t (7) ? × B = ??J + ?????E/?t (8) where E is the electric field vector, B is the magnetic field vector, ? is the scalar space charge, J is the current density vector, and ?? and ?? are scalar constants. Equation (5) is called Gauss's law, Eq. (7) is Faraday's law, and Eq. (8) is Ampère's law. Equation (6) is sometimes called Gauss's law for magnetism. (a) Write out Maxwell's equations in Cartesian tensor index notation. (b) Say there is no magnetic field, no space charge, and no current. Show that, if you take the electric field to be the gradient of an electric potential E = -??, the potential is governed by the Laplace equation. Show your work twice: once in vector form and once in tensor index form. (c) Say there is no electric field, no space charge, and no current. Show that, if you take the magnetic field to be the curl of the magnetic vector potential B = ? × A with ? · A = 0, the vector potential is governed by a vector Laplace equation. You may find Eq. (4) useful. Show your work twice: once in vector form and once in tensor index form. We'll see analogous equations for the velocity potential and the vector potential (generalized stream function) when we get to potential flow.
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In this convention, repeated indices are summed over. Maxwell's equations in index notation are: Show more…
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A.1 Find the magnetic field B produced by a long straight wire carrying a steady current I, as a function of I and the distance r from the wire. A.2 A physical electric dipole lying along the z-axis consists of charge +q at z = d/2 and charge -q at z = -d/2. Calculate the electrostatic potential φ(x) for large distances |x| ≫ d, as a function of q, d, |x|, and the angle θ between x and the z-axis. Show that the leading term for |x| ≫ d has the dipole potential form: φ_dip = (1/4πε_0) * (p · x̂ / x²) where p = qdẑ is the electric dipole moment. Using E = -∇φ, show that the electrostatic field due to the potential in (ii) is given by: E_dip(x) = (1/4πε_0) * (3(p · x̂)x̂ - p) / |x|³ A.3 Consider the Maxwell equations in vacuum: ∇ · E = ρ/ε_0 ∇ · B = 0 ∇ ∧ E = -∂B/∂t ∇ ∧ B = μ_0J + μ_0ε_0∂E/∂t Show that the two homogeneous Maxwell equations are automatically solved by the introduction of the electric and magnetic potentials φ and A via E = -∇φ - ∂A/∂t, B = ∇ ∧ A. Write down the two inhomogeneous Maxwell equations by eliminating E, B in favor of φ, A. Show that in the Lorentz gauge (c⁻²∂φ/∂t + ∇ · A = 0), the inhomogeneous equations take the form: ◻φ = -ρ/ε_0 ◸A = -μ_0J (◻ ≡ ∇² - (1/c²)∂²/∂t²)
Adi S.
Maxwell's equations for electromagnetism in free space can be written as follows: (i) ∇ · B = 0 (ii) ∇ · E = 0 (iii) ∇ × E + ∂B/∂t = 0 (iv) ∇ × B - (1/c²)∂E/∂t = 0 (a) The vector potential A is defined by B = ∇ × A and the scalar field ϕ by E = -∇ϕ - ∂A/∂t. Show that these relations are consistent with Maxwell's equations (i.e., that they satisfy Eqs. (i) and (iii)). (b) By choosing the Lorentz gauge ∇ · A + (1/c²)∂ϕ/∂t = 0, show that ϕ satisfies the wave equation ∇²ϕ - (1/c²)∂²ϕ/∂t² = 0. Hint: Substitute for E into (ii) and differentiate (v) with respect to time to eliminate A. (c) Show that the vector potential A also satisfies the wave equation. Hint: Substitute A and ϕ into the expressions for B and E in (iv). Then use the gradient of (v) to simplify the resulting expression.
i need help with this question: part 3
Lien L.
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