Question

a) Derive from the definition of the clectric current I = [s df - j through the arca S the dwhere dQ is the charge formula for the electric current inside a conducting wire I which flows through the cross section of the wire in the time dt stationary charge distribution, 2n wire have to be equal and independent of the cross sections of the wire both ends of the b) Show that for a 0, the currents on OSS c) Given a stationary charge distribution. Derive Kirchhoff's current law, which states that the sum of all inflowing currents is equal to the sum of all out flowing currents d) Show that a stationary field Ē in a volume V performs the power P = Jyj-EdV. Show that P UI holds for the case of a infinitesimal thin wire. Hint: P where W q frE-dr

Answer 1

"When the flow of charge distributed throughout a three-dimensional region. We describe it by volume current density. Consider a ""tube"" of infinitesimal cross section d a_L, running parallel to the flow. If the current in this tube is dI, the volume current density is I identicalto d I/ da_L J = rho v implies d I = rho V da_L I = integral rho v d a_L = d/dt (theta) implies (Basic equation) Now, I = d theta/dt = integral_S J middot d a_L \r\n nabla times B magnetic field = mu_0 I + mu_0 epsilon_0 partial differential E/ partial differential t electric field implies for stationary field middot partial differential E/partial differential t = 0 nabla times B = mu_0 I nabla middot (nabla downarrow 0 times B) = delta middot (mu_0 J) 0 = mu_0 (delta middot J) implies integral_v (delta^bar middot J^bar) dv = surfaceintegral_partial differential V J^bar ds = sigma_i doubleintegral_partial differential V J_i ds implies 0 = sigma_i (J_i doubleintegral_partial differential v_i ds) = sigma_i (J_i S_i) "
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