12/3/2004 section 9_2 Faraday's Law of Induction 1/2 Jim Stiles The Univ. of Kansas Dept. of EECS 9-2 Faraday’s Law of Induction Reading Assignment: pp. 277-286 Now let’s consider time-varying fields! Specifically, we consider what occurs when a magnetic flux density is not a constant with time (i.e., ()r,tB). Æ Maxwell’s Equations “recouple”, so that the electric field and magnetic flux density are related. Specifically, a time varying magnetic field is the source of a new, solenoidal electric field! HO: Faraday’s Law 9-2-1 Time-Varying Fields in Stationary Circuits Faraday’s Law is the basis for electric power generators! HO: The Electromotive Force12/3/2004 section 9_2 Faraday's Law of Induction 2/2 Jim Stiles The Univ. of Kansas Dept. of EECS Faraday’s Law is likewise the basis for the operation of transformers! HO: The Ideal Transformer HO: Eddy Currents12/3/2004 Faradays Law of Induction 1/2 Jim Stiles The Univ. of Kansas Dept. of EECS Faraday’s Law of Induction Say instead of a static magnetic flux density, we consider a time-varying B field (i.e., ()r,tB ). Recall that one of Maxwell’s equations is: ()()xr,trt∂∇=−∂BE Yikes! The curl of the electric field is therefore not zero if the magnetic flux density is time-varying! If the magnetic flux density is changing with time, the electric field will not be conservative! Q: What the heck does this equation mean ?!? A: Integrate both sides over some surface S: () ()xSSrds r,tdst∂∇⋅=− ⋅∂∫∫ ∫∫EB Applying Stoke’s Theorem, we get: () ()CSrd r,tdst∂⋅=− ⋅∂∫∫∫EBAv where C is the contour that surrounds the boundary of S.12/3/2004 Faradays Law of Induction 2/2 Jim Stiles The Univ. of Kansas Dept. of EECS Note that ()0Crd⋅≠∫E Av. This equation is called Faraday’s Law of Induction. Q: Again, what does this mean? A: It means that a time varying magnetic flux density ()r,tB can induce an electric field (and thus an electric potential difference)! Faraday’s Law describes the behavior of such devices such as generators, inductors, and transformers ! Michael Faraday (1791-1867), an English chemist and physicist, is shown here in an early daguerreotype holding a bar of glass he used in his 1845 experiments on the effects of a magnetic field on polarized light. Faraday is considered by many scientists to be the greatest experimentalist ever! (from “Famous Physicists and Astronomers” www.phy.hr/~dpaar/fizicari/index.html)12/3/2004 The Electromotive Force 1/4 Jim Stiles The Univ. of Kansas Dept. of EECS The Electromotive Force Consider a wire loop with surface area S, connected to a single resistor R. Since there is no voltage or current source in this circuit, both voltage v and current i are zero. Now consider the case where there is a time-varying magnetic flux density ()r,tB within the loop only. In other words, the magnetic flux density outside the loop is zero (i.e., ()0r,t=B outside of S). Say that this magnetic flux density is a constant with respect to position, and points in the direction normal to the surface S. In other words; ()()nˆr,t B t a=B S + v - i R12/3/2004 The Electromotive Force 2/4 Jim Stiles The Univ. of Kansas Dept. of EECS According to Faraday’s Law: () ()()()() ()()CSnCSbaab Srd r,tdstˆrd BtadstBtrd rd dst∂⋅=− ⋅∂∂⋅=− ⋅∂∂⋅+ ⋅=−∂∫∫∫∫∫∫∫∫ ∫∫EBEEEAAAAvv The contour from point a to point b is along a wire, which we presume to be a perfect conductor. Since the electric field within a perfect conductor is equal to zero, we find: ()0bard⋅=∫E A S + v - i R ()()Bnˆr,t t a=B C a b12/3/2004 The Electromotive Force 3/4 Jim Stiles The Univ. of Kansas Dept. of EECS Likewise, if we integrate through the resistor from point b to point a, we find: () ()abbard rd v⋅=− ⋅ =−∫∫EEAA Finally, we note that: Sds S=∫∫ where S is the surface area of the loop. Combining these results, we find: ()BtvSt∂=∂ Or, recalling that magnetic flux Φ is defined as: ()()Sr,t ds t⋅=Φ∫∫B we can write: ()()BtvSttt∂=∂∂Φ=∂12/3/2004 The Electromotive Force 4/4 Jim Stiles The Univ. of Kansas Dept. of EECS For this case, the voltage across the resistor is proportional to the time derivative of the total magnetic flux passing through the aperture formed by contour C. Using the circuit form of Ohm’s Law, we likewise find that the current in the circuit is: ()()1viRBtSRttRt=∂=∂∂Φ=∂ In other words, time-varying magnetic flux density can induce a voltage and current in a circuit, even though there are no voltage or current sources present! The voltage created is known as the electromotive force. The electromotive force is the basic phenomenon behind the behavior of: 1. Electric power generators 2. Transformers 3. Inductors12/3/2004 The Ideal Transformer 1/8 Jim Stiles The Univ. of Kansas Dept. of EECS The Ideal Transformer Consider the structure: * The “doughnut” is a ring made of magnetic material with very large relative permeability (i.e.,1rµ>> ). * On one side of the ring is a coil of wire with N1 turns. This could of wire forms a solenoid! * On the other side of the ring is another solenoid, consisting of a coil of N2 turns. This structure is an ideal transformer ! µ + _ ()1vt ()2vt+− 1N 2N ()1it LR ()2it12/3/2004 The Ideal Transformer 2/8 Jim Stiles The Univ. of Kansas Dept. of EECS * The solenoid on the left is the primary loop, where the one on the right is called the secondary loop. The current i1(t) in the primary generates a magnetic flux density ()r,tB . Recall for a solenoid, this flux density is approximately constant across the solenoid cross-section (i.e., with respect to r). Therefore, we find that the magnetic flux density within the solenoid can be written as: ()()r,t t=BB It turns out, since the permeability of the ring is very large, then this flux density will be contained almost entirely within the magnetic ring. + _ ()1vt ()2vt+− ()tB ()1it12/3/2004 The Ideal Transformer 3/8 Jim Stiles The Univ. of Kansas Dept. of EECS Therefore, we find that the magnetic flux density in the secondary solenoid is equal to that produced in the primary! Q: Does this mean also that ()()12vtvt=? A: Let’s apply Faraday’s Law and find out! Applying Faraday’s Law to the