Chapter 33 & 34 Review
Chapter 33: The Magnetic Field
Cyclotron Motion Newton’s Law Both a and F directed toward center of circle Equate
What is the force on those N electrons? Force on a single electron Force on N electrons Now use Lets make l a vector points parallel to the wire in the direction of the current when I is positive Force on wire segment:
Comments: Force is perpendicular to both B and l Force is proportional to I, B, and length of line segment Superposition: To find the total force on a wire you must break it into segments and sum up the contributions from each segment
Electric Field Magnetic Field q What are the magnitudes and directions of the electric and magnetic fields at this point? Assume q > 0 r Comparisons: both go like r -2, are proportional to q, have 4 in the denominator, have funny Greek letters Differences: E along r, B perpendicular to r and v
Magnetic Field due to a current Magnetic Field due to a single charge If many charges use superposition #2 q 2, v 2 #1 q 1, v 1 #3 q 3, v 3 r3r3 r2r2 r1r1 Where I want to know what B is
For moving charges in a wire, first sum over charges in each segment, then sum over segments
Summing over segments - integrating along curve I r Integral expression looks simple but…..you have to keep track of two position vectors Biot Savart law which is where you want to know B which is the location of the line segment that is contributing to B. This is what you integrate over.
Magnetic field due to an infinitely long wire x z I Current I flows along z axis I want to find B at the point I will sum over segments at points
r compare with E-field for a line charge
Gauss’ Law: Biot-Savart Law implies Gauss’ Law and Amperes Law But also, Gauss’ law and Ampere’s Law imply the Biot - Savart law Ampere’s Law
Electric field due to a single charge Magnetic field due to a single loop of current Guassian surfaces
# turns per unit length
Gauss’ Law: Biot-Savart Law implies Gauss’ Law and Amperes Law But also, Gauss’ law and Ampere’s Law imply the Biot - Savart law Ampere’s Law
Chapter 34: Faraday’s Law of Induction
Faraday’s Law for Moving Loops
Magnetic Flux Some surface Remember for a closed surface Magnetic flux measures how much magnetic field passes through a given surface Open surface Closed surface
Suppose the rectangle is oriented do that are parallel Rectangular surface in a constant magnetic field. Flux depends on orientation of surface relative to direction of B
Lenz’s Law In a loop through which there is a change in magnetic flux, and EMF is induced that tends to resist the change in flux What is the direction of the magnetic field made by the current I? A.Into the page B. Out of the page
Reasons Flux Through a Loop Can Change A.Location of loop can change B.Shape of loop can change C.Orientation of loop can change D.Magnetic field can change
Faraday’s Law for Moving Loops Faraday’s Law for Stationary Loops
L VBVB R VRVR VLVL I I Now I have cleaned things up making use of I B =-I R, I R =I L =I. Now use device laws: V R = RI V L = L dI/dt KVL: V L + V R - V B = 0 This is a differential equation that determines I(t). Need an initial condition I(0)=0
Solution: Let’s verify This is called the “L over R” time. Current starts at zero Approaches a value V B /R
What is the voltage across the resistor and the inductor?
L VBVB R VRVR VLVL I I Initially I is small and V R is small. All of V B falls across the inductor, V L =V B. Inductor acts like an open circuit. Time asymptotically I stops changing and V L is small. All of V B falls across the resistor, V R =V B. I=V B /R Inductor acts like an short circuit.
Let’s take a special case of no current initially flowing through the inductor Solution A: B: Initial charge on capacitor
Current through Inductor and Energy Stored Energy t
+
I V2V2 V1V1 Foolproof sign convention for two terminal devices 1.Label current going in one terminal (your choice). 2.Define voltage to be potential at that terminal wrt the other terminal V= V 2 -V 1 3. Then no minus signs Power to device KVL Loop Contribution to voltage sum = +V