Dr. Cherdsak Bootjomchai (Dr.Per) General Physics II By Dr. Cherdsak Bootjomchai (Dr.Per)

Sources of the Magnetic Field Chapter 8 Sources of the Magnetic Field

Force on or I in the filed A whole picture helps Charge q as source Current I as source Gauss’s Law Ampere’s Law Faraday’s Law Electric field E Magnetic field B Ampere-Maxwell Law Force on q in the field Force on or I in the filed Summarized in Lorentz force Maxwell equations

Math -- review Vector cross product: Magnitude of the vector : θ θ Determine the direction. If The right-hand rule: Four fingers follow the first vector. Bend towards the second vector. Thumb points to the resultant vector.

Sources of Electric field, magnetic field From Coulomb's Law, a point charge dq generates electric field distance away from the source: From Biot-Savart's Law, a point current generates magnetic field distance away from the source: Difference: A current segment , not a point charge dq, hence a vector. Cross product of two vectors, is determined by the right-hand rule, not .

Total Magnetic Field is the field created by the current in the length segment , a vector that takes the direction of the current. To find the total field, sum up the contributions from all the current elements The integral is over the entire current distribution mo = 4p x 10-7 T. m / A , is a constant of nature, the permeability of free space

Example: from a Long, Straight Conductor with current I The thin, straight wire is carrying a constant current I Construct the coordinate system and place the wire along the x-axis, and point P in the X-Y plane. Integrating over all the current elements gives

The math part

Now make the wire infinitely long The statement “the conductor is an infinitely long, straight wire” is translated into: q1 = p/2 and q2 = -p/2 Then the magnitude of the field becomes The direction of the field is determined by the right-hand rule. Also see the next slide.

For a long, straight conductor the magnetic field goes in circles The magnetic field lines are circles concentric with the wire The field lines lie in planes perpendicular to to wire The magnitude of the field is constant on any circle of radius a A different and more convenient right-hand rule for determining the direction of the field is shown

Example: at the center of a circular loop of wire with current I Y X From Biot-Savart Law, the field at O from is This is the field at the center of the loop Off center points are not so easy to calculate.

Magnetic Field of a Solenoid A solenoid is a long wire wound in the form of a helix. Each loop produces a magnetic field that adds together to form the total field. A reasonably uniform magnetic field can be produced in the space surrounded by the turns of the wire The field lines in the interior are nearly parallel to each other uniformly distributed The interior field of the solenoid is useful, ex. NMR imaging. Outside the solenoid, the field is weak as field lines from neighboring wires tend to cancel.

Magnetic Force Between Two Parallel Conductors Two parallel wires each carry steady currents The field due to the current in wire 2 exerts a force on wire 1 of F1 = I1ℓ B2 Substituting the equation for gives Check with right-hand rule: same direction currents attract each other opposite directions currents repel each other The force per unit length on the wire is And this formula defines the current unit Ampere.

Definition of the Ampere and the Coulomb The force between two parallel wires is used to define the ampere When the magnitude of the force per unit length between two long, parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A The SI unit of charge, the coulomb, is defined in terms of the ampere When a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 second is 1 C

Ampere’s Law Connects B with I Ampere’s law states that the line integral of around any closed path equals moI where I is the total steady current passing through any surface bounded by the closed path: This is a line integral over a vector field I

from a long, straight conductor re-calculated using Ampere’s Law Choose the Gauss’s Surface, oops, not again! Choose the Ampere’s loop, as a circle with radius a. Ampere’s Law says is parallel with , so

When the wire has a size: with radius R Outside of the wire, r > R Inside the wire, we need I’, the current inside the ampere’s circle

Plot the results The field is proportional to r inside the wire The field varies as 1/r outside the wire Both equations are equal at r = R

Ideal (infinitely long) Solenoid An ideal solenoid is approached when: the turns are closely spaced the length is much greater than the radius of the turns Apply Ampere’s Law to loop 2: n = N / ℓ is the number of turns per unit length

Magnetic Field of a Toroid The toroid has N turns of wire Find the field at a point at distance r from the center of the toroid (loop 1) There is no field outside the coil (see loop 2)

Magnetic Flux, defined the same way as any other vector flux, like the electric flux, the water (velocity) flux The magnetic flux over a surface area associated with a magnetic field is defined as The unit of magnetic flux is T.m2 = Wb (weber)

Magnetic Flux Through a Plane A special case is when a plane of area A, its direction makes an angle q with The magnetic flux is FB = BA cos q When the field is perpendicular to the plane, F = 0 (figure a) When the field is parallel to the plane, F = BA (figure b)

Postscript: Gauss’ Law in Magnetism Magnetic field lines do not begin or end at any point (current sources, no monopoles), they form closed loops. Gauss’ law in magnetism says the net magnetic flux through any closed surface is always zero, i.e. there can be no enclosed monopoles

Example problems A long, straight wire carries current I. A right-angle bend is made in the middle of the wire. The bend forms an arc of a circle of radius r. Determine the magnetic filed at the center of the arc. Formula to use: Biot-Savart’s Law, or more specifically the results from the discussed two examples: For the straight section For the arc The final answer: magnitude direction pointing into the page.

Another step toward Maxwell’s Equations Displacement Current Another step toward Maxwell’s Equations

Recall Ampere’s Law

òB.dl = 0Ienclosed ®problems |B| = 0I/2pR Measured |B| |B| = 0I/2pR |B| = 0/2pR = 0 I I=0 I Displacement current is a necessary evil. Let’s be really sad and lonely and play with Ampere’s Law near a capacitor. If we place the contour for integration around the wire, all is well – a magnetic field is measured and its magnitude is given by òH.dl. However, if we slide the contour for integration along the wire and pasty the capacitor, bad things happen. The measured field wil be continuous, òH.dl will be zero, as no current is enclosed. This is clearly nonsense. Something is going on inside the capacitor that supports the magnetic field near the plates. What is it?

Imagine a wire connected to a charging or discharging capacitor Imagine a wire connected to a charging or discharging capacitor. The area in the Amperian loop could be stretched into the open region of the capacitor. In this case there would be current passing through the loop, but not through the area bounded by the loop.

òB.dl = 0òJc.ds + 0òJd.ds |B| = 0I/2pR |B| = 0I/2pR |B| = 0I/2pR Id = òJd.ds I I If we include displacement current in the gap, then the paradox in Ampere’s Law is removed. We now have all 4 of Maxwell’s Equations …

If Ampere’s Law still holds, there must be a magnetic field generated by the changing E-field between the plates. This induced B-field makes it look like there is a current (call it the displacement current) passing through the plates.

Properties of the Displacement Current For regions between the plates but at radius larger than the plates, the B-field would be identical to that at an equal distance from the wire. For regions between the plates, but at radius less than the plates, the Ienc would be determined as through the total I were flowing uniformly between the plates.

Equation for Displacement current

Modified Ampere’s Law (Ampere-Maxwell Law)

Sources of the Magnetic Field The end of Chapter 8 Sources of the Magnetic Field