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Important!) Suppose that in a certain region of empty space, Ex = Ey = 0 and E₂ = Eot²/T2, where t is time, Eo is a constant with units of N/C, and T' is a constant with units of seconds. Suppose also that boundary conditions (and/or symmetry) imply that By = B₂ = 0 everywhere and that Bx = 0 on the y = 0 plane. Our goal is to determine how By depends on time t and on position coordinates x, y, z (which then completely determines in this region). (a) What is the magnitude and direction of the magnetic curl V x B in this region? (b) Show that Gauss's law for the magnetic field implies that Bx cannot depend on x (that is, that dBx/dx = 0). (c) Which component of the Ampere-Maxwell law implies that Bx cannot depend on z, and why? (d) Now that we know that Bx can depend at most on t and y, use a different component of the Ampere-Maxwell law to show that Bx = -2μo co (Eo/T²) ty.

Important!) Suppose that in a certain region of empty space, Ex = Ey = 0 and E₂ = Eot²/T2, where t is time, Eo is a constant with units of N/C, and T' is a constant with units of seconds. Suppose also that boundary conditions (and/or symmetry) imply that By = B₂ = 0 everywhere and that Bx = 0 on the y = 0 plane. Our goal is to determine how By depends on time t and on position coordinates x, y, z (which then completely determines in this region). (a) What is the magnitude and direction of the magnetic curl V x B in this region? (b) Show that Gauss's law for the magnetic field implies that Bx cannot depend on x (that is, that dBx/dx = 0). (c) Which component of the Ampere-Maxwell law implies that Bx cannot depend on z, and why? (d) Now that we know that Bx can depend at most on t and y, use a different component of the Ampere-Maxwell law to show that Bx = -2μo co (Eo/T²) ty.

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= Important!) Suppose that in a certain region of empty space, Ex Ey = 0 and E₂ = Eot²/T2, where t is time, Eo is a constant with units of N/C, and Tis a constant with units of seconds. Suppose also that boundary conditions (and/or symmetry) imply that By = B₂ = 0 everywhere and that Bx = 0 on the y = 0 plane. Our goal is to determine how By depends on time t and on position coordinates x, y, z (which then completely determines in this region). (a) What is the magnitude and direction of the magnetic curl V x B in this region? (b) Show that Gauss's law for the magnetic field implies that B, cannot depend on x (that is, that dBx/dx = 0). (c) Which component of the Ampere-Maxwell law implies that Bx cannot depend on z, and why? (d) Now that we know that Bx can depend at most on t and y, use a different component of the Ampere-Maxwell law to show that Bx = -2μo co (Eo/T²) ty.
Transcribed Image Text:= Important!) Suppose that in a certain region of empty space, Ex Ey = 0 and E₂ = Eot²/T2, where t is time, Eo is a constant with units of N/C, and Tis a constant with units of seconds. Suppose also that boundary conditions (and/or symmetry) imply that By = B₂ = 0 everywhere and that Bx = 0 on the y = 0 plane. Our goal is to determine how By depends on time t and on position coordinates x, y, z (which then completely determines in this region). (a) What is the magnitude and direction of the magnetic curl V x B in this region? (b) Show that Gauss's law for the magnetic field implies that B, cannot depend on x (that is, that dBx/dx = 0). (c) Which component of the Ampere-Maxwell law implies that Bx cannot depend on z, and why? (d) Now that we know that Bx can depend at most on t and y, use a different component of the Ampere-Maxwell law to show that Bx = -2μo co (Eo/T²) ty.
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