SLIDE 1
Solenoid
B I L
Solenoid Solenoid L I B If n = number of turns per unit length - - PDF document
Solenoid Solenoid L I B If n = number of turns per unit length - - PDF document
Solenoid Solenoid L I B If n = number of turns per unit length B d s B L Amperes Law: B d s I B L (nL) I 0 0 B n I 0 Note
SLIDE 2
SLIDE 3
Toroid
r 2 B s d B
I r 2 N B I r) 2 (n r 2 B I s d B
Ampere’s Law: N = Total number of turns
SLIDE 4
Structure of Equations
E B
Interaction between charges Interaction between moving charges/ currents Coulomb’s Law Biot‐Savart Law Gauss’s Law Gauss’s Law (Conceptual) Gauss’s Law Ampere’s Law (Calculation) Parallel capacitor gives uniform E field Solenoid gives uniform B field
r ˆ r dq 4 1 E d
2
r r ˆ s d 4 I B d
2
in
q A d E
A d B
in
q A d E
in 0 I
s d B
nI B E
SLIDE 5
Maxwell’s Equations
Maxwell’s equations describe all the properties of electric and magnetic fields and there are four equations in it: Integral form Differential form (optional)
Name of equation
1st Equation Electric Gauss’s Law Magnetic Gauss’s Law Ampere’s Law (Incomplete)
enclosed
Q A d E
A d B
E B
Not yet2nd
Equation 3rd Equation
Lorentz force equation is not part of Maxwell’s equations. It describes what happens when charges are put in an electric or magnetic fields:
) B v E (q F I d B
enclosed
J B
SLIDE 6
Class 35: Faraday’s Law Part I – Maxwell’s 4th Equation
SLIDE 7
B
Imaginary loop in an electric and magnetic field
We will do two types of integrals for the closed loop: 1. Magnetic flux Note that B0 (Maxwell’s 2nd equation) because this is not a 3 dimensional closed surface. 2. Electromotive force (emf, ) loop = 0 for electrostatic case. Note that loop = 0 does not mean E =0.
A d B
B
s d E
loop loop
E (non-uniform)
SLIDE 8
Example
What is the magnetic flux through the rectangular loop?
I d a b
SLIDE 9
Electric Potential V A B
If E(r) is conservative, the potential difference V is defined as the negative work done by the force F(r) (which is path independent), divided by the charge (of the test charge).
1 ` 1 ` 1 `
r d ) r ( F
- U
f i
Pay attention to the negative sign
r d ) r ( E
- q
U V
f i
Unit of electric potential = J/C =V
Old slide from class 13 V=0 for closed loop
SLIDE 10
Warning
In the discussion here we will assume electric (force) field is a conservative (force) field. This will not be the case if there is a changing magnetic field. We will come to this point later in the semester.
Old slide from class 13
SLIDE 11
E B
Faraday’s Law – Part 1 (Maxwell’s 4th equation)
Notes:
- 1. We find a new way to produce an
electric field.
- 2. loop of electric fields produced
this way does not equal to 0.
A d B t s d E
- r
t
loop B loop