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Unformatted text preview: Physics 1C – First Week Class notes : Walter Gekelman Winter 2008 1 Physics 1C Winter 2008 Class Notes : Walter Gekelman Let us review once again the Force laws as well as those involving electric forces: (1) ! F = q ! E + ! v ! ! B ( ) Force law with Lorentz term (2) ! ! E i ˆ ndA = Q " Gauss’s law Magnetic Fields : (3) ! B i ˆ ndA = ! no magnetic monopoles and Ampere’s Law: (4) ! B i d ! l = μ " ! i . This relates the magnetic field to the current. Note the integral sign refers to the integral around a closed loop. Magnetic fields, in turn, always come from moving charges, or electrical current which is a collection of moving charges. For a moving charge: ( 5) ! B = μ 4 ! q ! v " ˆ r ( ) r 2 here r is the distance to the observer of the magnetic field and v is the velocity of the charge. Because there is a cross product in the expression the magnetic field is out of the plane containing ! r , ! v ( ) . This is all well illustrated in Figure 28.1 in the text. Fig 281 B from a moving charge μ = 4 ! " 10 # 7 N/A 2 (Newton's/Amp 2 ) Physics 1C – First Week Class notes : Walter Gekelman Winter 2008 2 What about the magnetic field from a wire with an arbitrary shape? To this end we use the Law of Biot and Savart: (6) ! B r ( ) = μ o i 4 ! d ! l " ˆ r r 2 # Always works, no symmetry required Note d ! B r ( ) = μ o i 4 ! d ! l " ˆ r r 2 (from a segment dl of the wire) Equations 5 and 6 are basically the same. Consider a current moving through a tiny segment of wire of length dl. The current is constrained to be within the wire which implies that Id ! l = ! Idl . The definition of current density ! J (current per unit area ) and current (I) is: (7) ! J = qn ! v I = ! J i d ! A ! = qn ! v i d ! A ! Here j is the current density and the dot product is between the velocity of the charge and the normal to the area. If the current flows into an area (along the normal to it) and not at an angle then ! I = nq ! vA . Putting this into the Biot Savart Law d ! B = μ 4 ! ( nqdlA ) ! v " ! r r 2 . n is thenumber of charges per volume. The volume of a segment of the wire is Adl and if there is only 1 charge the equation reduces to equation 5. Let us do a problem with the law of Biot and Savart. What is the magnetic field (on axis) above a circular ring carrying a current I? The current I is counterclockwise dl r B Physics 1C – First Week Class notes : Walter Gekelman Winter 2008 3 dB = μ Idl 4 ! r 2 note d ! l " ˆ r =dl observer Physics 1C – First Week Class notes : Walter Gekelman Winter 2008 4 The answer (given on page 720 is) B = μ IR 2 2 R 2 + x 2 ( ) 3/ 2 ˆ i x dB x dB y dB θ θ R r 2 2 cos x R B dB dB x R ! = = + " " Bx Physics 1C – First Week Class notes : Walter Gekelman Winter 2008 5 Now here’s where you need to use the law of Biot and Savart: The gold structures are magnetic coils for a type of machine called a Stellerator. The pink “stuff” in the center is a, plasma which is supposed to be confined by these magnetic...
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 Winter '07
 Whitten
 Force, Magnetic Field, Walter Gekelman Winter, Walter Gekelman, irst Week Class

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