Chapter 8 Magnetic Fields: I Magnetic induction a The divergence of b Magnetic monopoles The Vector potentiala a and v Integral of a over a Closed Curve
Chapter 8 Magnetic Fields:I ◼ Magnetic Induction ◼ The Divergence of B ◼ Magnetic Monopoles ◼ The Vector Potential A ◼ A and V ◼ Integral of A over a Closed Curve
8. 1 The Magnetic Induction B See fig 8-1 a circuit carrying a current I generates the magnetic induction at the point Pas B X ri 4丌 where the integration is carried out over the circuit c and the constant uo is called the permeability of free space 10=4丌×10-7 tesla meter/ ampere Remark: the unit vector r1 points from the source to the point of observation i.e. from di to P. b has unit tesla
dB dI figure 8-l The magnetic induction dB=(Ho/4a)l dI x r/ produced by an element i dl of the current I in a circuit
If the current is distributed in space with a current den sity j, then the element of current Idl is replaced by dr and the magnetic induction expressed as r B 4丌 l-2 ere the integration is carried over the volume T'occu pied by the currents Remarks: These two expressions for b are equivalent The flux of the magnetic induction B through a surface s is defined as Φ=/sB·da Φ has unit weber
Example: Long Straight Wire Carrying a Current See Fig 8-3 dB dI Figure 8-3 The magnetic induction dB produced by an element i dl of the current I in a long straight wire
Example: Long Straight Wire Carrying a Current See Fig. 8-3
An element dl with current I produces a magnetic in duction dB= 0,dl×r1p0,sin6dl 4丌 4丌 where 0 is the angle between dl and rI, and 1 is the unit vector in the azimuthal direction Now using sin Adl= cos adl=rda and p=r cos a, sin odl rda da cos ada 2 Thus integration yields the magnetic induction B=4r//2 cos ada 4丌-m2p d12丌 See Fig 8-4
( Figure 8-4 Lines of B in a plane perpendicular to a long straight wire carrying a current I. The density of the lines is inversely proportional to the distance to the wire. Lines close to the wire are not shown
Example 2 Circular Loop Magnetic Dipole Moment m See fig. 8-5 for a loop of radius a wit a current l We want to know the component of b along the z axis produced by the loop db- bo, dl X I1, dBi 4T r2 COS 4丌 B Z 4丌 By cos 0= a/r, and r=z+a, we have 22 a 2)1/2
4B Figure 8-5 The magnetic induction dB produced by an element I di at a point on the axis of circular current loop of radius a. The projection of dB on the axis is dB
Figure 8-6 Lines of B in a plane containing the axis of the loop of Fig. 85. The direction of the current in the loop is shown by the dot and the cross