Electricity & Magnetism Lecture 2: Electric Fields
Today’s Concepts: A) The Electric Field B) Con nuous Charge Distribu ons
Electricity & Magne sm Lecture 2, Slide 1 Your Comments
Suddenly, terrible haiku: Posi ve test charge Repelled by op mists Cup half electric a li le bit concentra on it requires.
I had problems understanding the concept and calculation of charge on and infinite line
??Infinite lines of chargee??? it would help alot to go over those, everything else is simmplle
Posi ves and nega ves, what do they really mean? It seems like we, as humans, just decided to name them this.
define Electric field and electric force and charge in words please
I almost forgot how exciting smartphysics was, yay
Irony?
Electricity & Magne sm Lecture 2, Slide 2 Coulomb’s Law (from last time)
If there are more than two charges present, the total force on any given charge is just the vector sum of the forces due to each of the other charges: q2 q2
F F4,1 4,1 F F2,1 2,1 q1 q1 F F1 1 F F3,1 3,1 q4 q4
F q3 F1 q3 1 F F2,1 2,1 F F3,1 3,1 +q1 −> −q1 Þ direc on reversed F F4,1 MATH: 4,1 =
Electricity & Magne sm Lecture 2, Slide 3 In the diagrams below, the magnitude and direc on of the electric field is represented by the length and direc on of the blue arrows. Which of the diagrams best represents the electric field from a nega ve charge? 3 charges are at the corners of an equilateral triangle Where is the E field 0? The black line is 1/2 way between the base and the top charge. A.Above the black line B. Below the black line C. on the black line D.nowhere except infinity E. somewhere else
! Electric Field Line Applet
Electric Field PhET
Electric Field “What exactly does the electric field that we calculate mean/represent? “ “What is the essence of an electric field? “ The electric field E at a point in space is simply the force per unit charge at that point.
Electric field due to a point charged par cle
q2 Superposi on E4
E2 E Field points toward nega ve and E3 Away from posi ve charges. q4
q3
Electricity & Magne sm Lecture 2, Slide 4 CheckPoint: Electric Fields1
A
x B +Q −Q
Two equal, but opposite charges are placed on the x axis. The posi ve charge is placed to the le of the origin and the nega ve charge is placed to the right, as shown in the figure above.
What is the direc on of the electric field at point A? o Up o Down o Le o Right o Zero
Electricity & Magne sm Lecture 2, Slide 5 CheckPoint Results: Electric Fields1
A
x B +Q −Q
Two equal, but opposite charges are placed on the x axis. The posi ve charge is placed to the le of the origin and the nega ve charge is placed to the right, as shown in the figure above.
What is the direc on of the electric field at point A? o Up o Down o Le o Right o Zero
Electricity & Magne sm Lecture 2, Slide 6 CheckPoint: Electric Fields2
A
x B +Q −Q
Two equal, but opposite charges are placed on the x axis. The posi ve charge is placed to the le of the origin and the nega ve charge is placed to the right, as shown in the figure above.
What is the direc on of the electric field at point B? o Up o Down o Le o Right o Zero
Electricity & Magne sm Lecture 2, Slide 7 CheckPoint Results: Electric Fields2
A
x B +Q −Q
Two equal, but opposite charges are placed on the x axis. The posi ve charge is placed to the le of the origin and the nega ve charge is placed to the right, as shown in the figure above.
What is the direc on of the electric field at point B? o Up o Down o Le o Right o Zero Polarization Demo
Electricity & Magne sm Lecture 2, Slide 8 CheckPoint: Magnitude of Field (2 Charges)
In which of the two cases shown below is the magnitude of the electric field at the point labeled A the largest?
A +Q +Q A
−Q +Q
Case 1 Case 2 o Case 1 o Case 2 o Equal
Electricity & Magne sm Lecture 2, Slide 9 CheckPoint Results: Magnitude of Field (2 Chrg)
In which of the two cases shown below is the magnitude of the electric field at the point labeled A the largest?
E A +Q +Q A E
−Q +Q
Case 1 Case 2
“The upper le +Q only affects the x direc on in both and the lower o Case 1 right (+/-)Q only affects the y direc on so in both, nothing cancels o Case 2 out, so they'll have the same magnitude. ” o Equal
Electricity & Magne sm Lecture 2, Slide 10 CheckPoint: Magnitude of Field (2 Charges)
In which of the two cases shown below is the magnitude of the electric field at the point labeled B the largest?
A +Q +Q B B A
−Q +Q
Case 1 Case 2 o Case 1 o Case 2 o Equal
Electricity & Magne sm Lecture 2, Slide 9 Clicker Question: Two Charges
Two charges q1 and q2 are fixed at points (-a,0) and (a,0) as shown. Together they produce an electric field at point (0,d) which is directed along the nega ve y-axis.
y (0,d)
q1 q2 (−a,0) (a,0) x
Which of the following statements is true:
A) Both charges are nega ve B) Both charges are posi ve C) The charges are opposite D) There is not enough informa on to tell how the charges are related
Electricity & Magne sm Lecture 2, Slide 11 + + _ _
+ _
Electricity & Magne sm Lecture 2, Slide 12 CheckPoint Results: Motion of Test Charge
A posi ve test charge q is released from rest at distance r away from a charge of +Q and a distance 2r away from a charge of +2Q. How will the test charge move immediately a er being released?
“The force is propor onal to the charge divided by o To the le the square of the distance. Therefore, the force of o To the right the 2Q charge is 1/2 as much as the force of the Q charge. “ o Stay s ll “Even though the charge on the right is larger, it is o Other twice as far away, which makes the force it exherts on the test charge half that as the charge on the le , causing the charge to move to the right.”
The ra o between the R and Q on both sides is 1:1 meaning they will result in the same magnitude of electric field ac ng in opposite direc ons, causing q to remain s ll.
Electricity & Magne sm Lecture 2, Slide 13 Electric Field Example
+q P What is the direc on of the electric field d at point P, the unoccupied corner of the square? −q +q d Need to Need to B) E) A) C) D) know d know d & q
Calculate E at point P.
q q = ⇡ Ex k 2 2 cos 4 0d p2d 1 B C B C B q ⇣ q ⌘ C = @ ⇡ A Ey k 2 2 sin 4 0d p2d 1 B C B C @B ⇣ ⌘ AC Electricity & Magne sm Lecture 2, Slide 14 About rˆ y
ˆj ~r =r cos ✓ ˆi + r sin ✓ ˆj ˆi 12 12 12
x r12 cos ✓ ˆi + r12 sin ✓ ˆj 2 rˆ12 = r12 ˆ ˆ ~r12 rˆ12 = cos ✓ i + sin ✓ j θ 1
rˆ12 For example
~r12 =4ˆi + 3 ˆj 2 2 (5,5) r12 = p4 + 3 = 5 4 ~r12 cos ✓ = 5 θ 3 (1,2) sin ✓ = 5
r12 cos ✓ ˆi + r12 sin ✓ ˆj rˆ12 = r12 rˆ12 4 ˆ 3 ˆ rˆ12 = 5 i + 5 j Continuous Charge Distributions
“I don't understand the whole dq thing and lambda.” Summa on becomes an integral (be careful with vector nature)
WHAT DOES THIS MEAN ? Integrate over all charges (dq) r is vector from dq to the point at which E is defined
Linear Example: λ = Q/L dE pt for E r charges dq = λ dx
Electricity & Magne sm Lecture 2, Slide 15 Clicker Question: Charge Density “I would like to know more about the charge density.”
Some Geometry Linear (λ = Q/L) Coulombs/meter Surface (σ = Q/A) Coulombs/meter2 Volume (ρ = Q/V) Coulombs/meter3
What has more net charge?. A) A sphere w/ radius 2 meters and volume charge density ρ = 2 C/m3 B) A sphere w/ radius 2 meters and surface charge density σ = 2 C/m2 C) Both A) and B) have the same net charge.
Electricity & Magne sm Lecture 2, Slide 16 Prelecture Question
A) What is the electric field at point a?
B)
C)
D)
E) Clicker Question: Calculation y P “How is the integra on of dE over L worked out, step by step?” r h Charge is uniformly distributed along dq = λ dx the x-axis from the origin to x = a. The x charge density is λ C/m. What is the x- x component of the electric field at a point P: (x,y) = (a,h)? We know:
What is ?
A) B) C) D) E)
Electricity & Magne sm Lecture 2, Slide 19 Clicker Question: Calculation
Charge is uniformly distributed along the x-axis from the origin to x a. The θ2 = y P charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? r h
θ1 θ2 x x a dq = λ dx We know:
What is ?
A) B) C) D)
Electricity & Magne sm Lecture 2, Slide 20 Clicker Question: Calculation
Charge is uniformly distributed along the x-axis from the origin to x a. The θ2 = y P charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? r h
θ1 θ2 x x a We know: dq = λ dx
What is ?
a 1 dx dx ✓ k ✓ A) k cos 2 2 2 B) cos 2 2 2 (a x) + h 0 (a x) + h Z 1 Z C) none of the above cosθ2 DEPENDS ON x! Electricity & Magne sm Lecture 2, Slide 21 Clicker Question: Calculation
Charge is uniformly distributed along the x-axis from the origin to x a. The θ2 = y P charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? r h
θ1 θ2 x x a dq = λ dx We know:
What is ?
A) B) C) D)
Electricity & Magne sm Lecture 2, Slide 22 Calculation
Charge is uniformly distributed along the x-axis from the origin to x a. θ2 = y P The charge density is λ C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? r h
θ1 θ2 x x a We know: dq = λ dx
What is ? a a x h = Ex(P) = k 1 Ex(P) k dx pa2 + h2 0 (a x)2 + h2 ! Z p Electricity & Magne sm Lecture 2, Slide 23 Observation
Charge is uniformly distributed along the x-axis from the origin to x = a. θ y P 2 The charge density is λ C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? r h
θ1 θ2 Note that our result can be rewri en x x a more simply in terms of θ1. dq = λ dx
k h k Ex(P) = 1 Ex(P) = (1 sin ✓1) h p 2 2 h + a ! h
Exercise for student: k⇥ ⇤/2 Change variables: write x in terms of θ Ex(P) = d cos h 1 Result: obtain simple integral in θ Z
Electricity & Magne sm Lecture 2, Slide 24