Download Tutorial 11 Energy Flow

Name________________________________________ Energy Flow in a Simple Circuit 
A steady uniform current density J is flowing upwards through a cylindrical resistor (as shown in the diagram at right). The resistor is made of a poorly conducting material with high resistivity ρ = 1 σ , where σ is the conductivity of the material. A. This next diagram shows the same resistor in cross-­‐section. At the point indicated inside the resistor (circular dot), determine the direction of both 

the electric field E and the magnetic field B , and sketch these vectors at that point. At that same point, draw an arrow indicating the direction of the Poynting vector:  
 E×B
S=
µ0
Describe in words how energy is flowing into the resistor. 1 Name________________________________________ 2 B. Consider an Amperian loop (dashed lines) with length  and width w that straddles the surface of the resistor. Using Faraday’s Law:  
 
d
E
⋅
d

=
−
B
⋅ d a ∫
dt ∫∫
determine whether the value of the closed line integral of the electric field along this Amperian loop is positive, negative or zero. (Be sure to indicate your direction of integration):  
Sign of 
∫ E ⋅ d = Explain your reasoning. Use this information to determine the relationship between the parallel components of the electric field ( E ) just inside and just outside the resistor. You may continue, but be sure to check your answer to Part A with an instructor. Name________________________________________ 3 C. The parallel components of the electric field ( E ) outside the resistor are shown inside the dashed region just to the left of the surface. Sketch in the diagram the direction of the perpendicular components of the electric field ( E⊥ ) in that same region. [Hint: What must they look  
E
∫ ⋅ d = 0 to be true around the closed loop surrounding the dashed region?] On the right side of the diagram, sketch what you think the total electric field lines should look like just outside the resistor. Is the volume charge density inside the resistor positive, negative or zero? Where are the charges located that are responsible for the perpendicular components of the electric field outside the resistor? like in order for Name________________________________________ 4 D. Suppose the steady state surface charge on the resistor is distributed as shown in the diagram below, where there is a greater buildup of positive charge toward the bottom, zero surface charge in the middle, and an increasingly negative surface charge toward the top (the charge varies smoothly from positive to negative). Sketch the magnitude and direction of the perpendicular components of the electric field ( E⊥ ) at the points indicated just outside and to the left of the resistor (circular dots). Suppose the resistor were instead made of an ideal conductor ( σ → ∞ ). Describe the parallel component of the electric field just outside this ideal conductor. [Hint: What happens to the electric field inside the 

material if the current density ( J = σ E ) remains finite as σ → ∞ ?] Name________________________________________ 5 E. The diagram below depicts ideal conducting wires ( σ → ∞ ) connecting an ideal battery with a resistor (R) made of a poorly conducting material. The surface charge buildup on the ideal conducting wires is roughly uniform, and varies smoothly from positive to negative on the surface of the resistor. [The diagram is schematic, so assume the nine circular dots shown in the diagram are lying just outside the surfaces of the circuit elements.] In the diagram above, sketch the total electric field vectors at the nine points indicated (circular dots, which are situated just outside the surfaces of the circuit elements). At the same nine points, indicate the direction of the magnetic field due to the current flowing through the circuit.   
Now sketch the direction of the Poynting vector ( S  E × B ) at each point. From this information, describe in words how energy is flowing from the battery to the resistor. Where is the energy flowing and in what direction(s)? Is energy flowing through the ideal conducting wires?