Ch5. 靜磁學(Magnetostatics) 5.1 羅倫茲力定律(The Lorentz Force Law) 5.2 必歐沙伐定律(The Biot-Savart Law) 5.3 磁場的散度與旋度(The Divergence and Curl of B) 5.4 磁向量位勢(Magnetic Vector Potential)

5.1.1 The Lorentz Force Law-磁場(Magnetic Fields) Bar magnet Conducting wire Parallel wires carrying currents in the same direction attract each other Parallel wires carrying currents in opposite direction repel each other

The Lorentz force 5.1.2.1 磁力(Magnetic Forces) Additional speed parallel to B

5.1.2.2 磁力(Magnetic Forces) z

5.1.2.3 磁力(Magnetic Forces) But the particle started from rest ( ), at the origin ( );

5.1.2.4 磁力(Magnetic Forces) This is the formula for a circle, of radius R, whose center (0, Rt, R) travels in the y-direction at a constant speed

5.1.3.1 電流(Currents) Example 5.3 : A rectangular loop of wire, supporting a mass m, hangs vertically with one end in a uniform magnetic field B, which points into the page in the shaded region. For what current I, in the loop, would the magnetic force upward exactly balance the gravitational force downward?

5.1.3.2 電流(Currents) What happens if we now increase the current? The loop rises a height of h quB qwB Rise the loop v u w

5.1.3.3 電流(Currents) Example 5.4 a) A current I is uniformly distributed over a wire of circular cross section, with radius a. Find the volume current density J. b) Suppose the current density in the wire is proportional to the distance from the axis, J = ks. Find the total current in the wire. The current in the shaded path

5.1.3.4 電流(Currents) The current crossing a surface S can be written as The total charge per unit time leaving a volume V is Because charge is conserved Continuity equation

5.2.1 穩定電流(Steady Currents) Continuity equation Actually, it is not necessary that the charges be stationary, but only that the charge density at each point be constant ( be independent with time).

5.2.2 穩定電流的磁場(The Magnetic Field of a Steady Current) Example 5.5 Find the magnetic field a distance s from a long straight wire carrying a steady current I s   I ℓ dℓ

5.2.3 穩定電流的磁場(The Magnetic Field of a Steady Current)  s 1  2 I ℓ dℓ

5.2.4 穩定電流的磁場(The Magnetic Field of a Steady Current) The field at 2 due to 1 is into the page The force on 2 due to 1 is The force per unit length

5.2.5 穩定電流的磁場(The Magnetic Field of a Steady Current) Example 5.6 Magnetic Field on the Axis of a Circular Current Loop

5.2.6 穩定電流的磁場(The Magnetic Field of a Steady Current) For surface and volume currents, the Biot-Savart Law becomes Problem 5.8 a) Find the magnetic field at the center of a square loop, which carries a steady current I. Let R be the distance from center to side. R for Four sides

5.2.7 穩定電流的磁場(The Magnetic Field of a Steady Current) b) Find the field at the center of a rectangular n-sided polygon, carring a steady current I. Let R be the distance from center to any side. for n sides c) Check that your formula reduces to the field at the center of a circular loop, in the limit n  ∞. n  ∞

5.2.8 穩定電流的磁場(The Magnetic Field of a Steady Current) Problem 5.9 Find the magnetic field at point P for each of the steady current configurations. The vertical and horizontal lines produce no field at P. I b The two quarter-circles a P The two half-lines are the same as one infinite line R I P Total: I The half-circle:

5.3.1.1 磁場的散度與旋度– 直線電流(Straight-Line Currents) The magnetic field of an infinite straight wire Notice that the answer is independent of s; that’s because B decreases at the same rate as the circumference increases. In fact, it doesn’t have to be a circle; any old loop that encloses the wire would give the same answer. For if we use cylindrical coordinates (s,,z), with this current flowing along the z axis, This assumes the loop encircles the wire exactly once.

5.3.1.2 磁場的散度與旋度– 直線電流(Straight-Line Currents) Suppose we have a bundle of straight wires. Each wire that posses through our loop contributes 0I, and those outside contribute nothing. I4 I3 I2 I enc : total current enclosed by the integration path I1 If the flow of charge is represented by a volume current density J, the enclosed current is Applying Stokes’ theorem

5.3.2.1 磁場的散度與旋度(The Divergence and Curl of B) (x,y,z) The Biot-Savart law for the general case of a volume current reads d’ (x’,y’,z’) Because J doesn’t depend on the unprimed variables (x,y,z) The divergence of the magnetic field is zero!

5.3.2.2 磁場的散度與旋度(The Divergence and Curl of B) = 0 = 0

5.3.2.3 磁場的散度與旋度(The Divergence and Curl of B) The x component For steady currents the divergence of J is zero This contribution to the integral can be written On the boundary J = 0

5.3.2.4 磁場的散度與旋度(The Divergence and Curl of B) Ampère’s law in differential form

5.3.3.1 安培定律的應用(Applications of Ampère’s Law) Ampère’s law in differential form Ampère’s law in integral form Example 5.8 Find the magnetic field of an infinite uniform surface current , flowing over the xy plane

5.3.3.2 安培定律的應用(Applications of Ampère’s Law) What is the direction of B? From the Biot-Savart law Could it have a z-component ?  no (symmetry) The magnetic field points to the left above the plane and to the right below it

5.3.3.3 安培定律的應用(Applications of Ampère’s Law) Example 5.9 Find the magnetic field of a very long solenoid, consisting of n closely wound turns per unit length on a cylinder of radius R and carrying a steady current I. Where N is the number of turns in the length

5.3.3.4 安培定律的應用(Applications of Ampère’s Law) Example 5.10 Find the magnetic field of a toroidal coil, consisting of a circular ring around which a long wire is wrapped.

5.3.3.5 安培定律的應用(Applications of Ampère’s Law) Problem 5.13 A steady current I flows down a long cylindrical wire of radius a. Find the magnetic field, both inside and outside the wire, if (a). The current is uniformly distributed over the outside surface of the wire. (b). The current is distributed in such a way that J is proportional to s, the distance from the axis. a I (a)

5.3.3.5 安培定律的應用(Applications of Ampère’s Law) (b) For s < a For s > a

5.3.4. 靜磁學與靜電學之比較(Comparison of Magnetostatics and Electrostatics) The divergence and curl of the electrostatic field are Gauss’s law The divergence and curl of the magnetostatic field are Ampère’s law “The electric force is stronger than the magnetic force. Only when both the source charge and the test charge are moving at velocities comparable to the speed of light, the magnetic force approaches the electric force.”

5.4.1.1. 磁向量位勢(Magnetic Vector Potential) V: electric scalar potential You can add to V any function whose gradient is zero : magnetic vector potential You can add to any function whose curl is zero We will prove that :

5.4.1.2. 磁向量位勢(Magnetic Vector Potential) Suppose that our original vector potential is not divergenceless Because we can add to any function whose curl is zero If a function  can be found that satisfies Mathematically identical to Poisson’s equation In particular, if  goes to zero at infinity, then the solution is

5.4.1.3. 磁向量位勢(Magnetic Vector Potential) By the same token, if goes to zero at infinity, then It is always possible to make the vector potential divergenceless so This again is a Poisson’s equation Assuming goes to zero at infinity, then For line and surface currents

5.4.1.4. 磁向量位勢(Magnetic Vector Potential) Example 5.11 A spherical shell, of radius R, carrying a uniform surface charge , is set spinning at angular velocity . Find the vector potential it produces at point . The integration is easier if we let lie on the z axis, so that  is titled at an angle .

5.4.1.5. 磁向量位勢(Magnetic Vector Potential) where Because We just consider

5.4.1.6. 磁向量位勢(Magnetic Vector Potential) Letting , the integral becomes If the point lies inside the sphere, Then R > r.  If the point lies outside the sphere, Then R < r. 

5.4.1.7. 磁向量位勢(Magnetic Vector Potential) If the point lies inside the sphere, Then R > r.  If the point lies outside the sphere, Then R < r.  Noting that For the point inside the sphere For the point outside the sphere

5.4.1.8. 磁向量位勢(Magnetic Vector Potential) We revert to the original coordinates, in which coincides with the z axis and the point is at (r,,) For the point inside the sphere The magnetic field inside this spherical shell is uniform! For the point outside the sphere

5.4.1.9. 磁向量位勢(Magnetic Vector Potential) Example 5.12 Find the vector potential of an infinite solenoid with n turns per unit length, radius R, and current I We cannot use because the current itself extends to infinity. Notice that Since the magnetic field is uniform inside the solenoid : For s < R For an amperian outside the solenoid For s > R

5.4.2.1. 總結;靜磁邊界條件(Summary; Magnetostatic Boundary Conditions) ?

5.4.2.2. 總結;靜磁邊界條件(Summary; Magnetostatic Boundary Conditions) For an amperian loop running perpendicular to the current Perpendicular to the current For an amperian loop running parallel to the current Parallel to the current Where is a unit vector perpendicular to the surface, pointing “upward”.

5.4.2.3. 總結;靜磁邊界條件(Summary; Magnetostatic Boundary Conditions) The vector potential is continuous across any boundary For an amperian loop of vanishing thickness

5.4.3.1. 向量位勢的多極展開(Multipole Expansion of the Vector Potential) A multipole expansion which is an approximate formula and valid at distant points) for the vector potential of a localized current distribution. I

5.4.3.2. 向量位勢的多極展開(Multipole Expansion of the Vector Potential) monopole dipole quadrupole The magnetic monopole tern is always zero Let , where is a constant vector.

5.4.3.3. 向量位勢的多極展開(Multipole Expansion of the Vector Potential) Let where

5.4.3.4. 向量位勢的多極展開(Multipole Expansion of the Vector Potential) Example 5.13 Find the magnetic dipole moment of the “Bookend-shaped” loop shown in Figure below. All sides have length w, and it carries a current I. w w w The wire could be considered the superposition of two plane square loops shown in Fig. 5.53 of text book. The combined (net) magnetic dipole moment is

5.4.3.5. 向量位勢的多極展開(Multipole Expansion of the Vector Potential) The magnetic dipole moment is independent of the choice of origin. The magnetic field of a (pure) dipole is easier to calculate if we put the dipole moment at the origin and let it point in the z-direction. This is identical in structure to the field of an electric dipole!