Superposition and Dipole E Field

Superposition Electric Field From a Dipole

Superposition and Dipole E~ ﬁeld

PHYS 272 - David Blasing

Tuesday June 11th

1 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Principle of Superposition

Definition: Superposition The net electric field at a position in space is the vector sum of every electric field made at that location by all the other charged particles around.

The electric field created by a charged particle is not affected by the presence of other charged particles or electric fields nearby.

2 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Superposition Example

3 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere ~ ~ Enet is related to Fnet

Once you have calculated the E~net due to all the other charges, then you can quickly calculate the force F~net = Q ∗ E~net on any amount of charge Q placed at that location

4 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Clicker Question 1

What is the direction of E~net at the location of q3 if |q2| ≈ 2|q1| ?

5 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Clicker Question 1

What is the direction of E~net at the location of q3?

6 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Clicker Question 2

7 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Clicker Question 3

8 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Clicker Question 4

9 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere Clicker Question 5

10 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere

Electric ﬁeld of a uniformly charged sphere

for r > R

~ 1 Q Esphere = 2 rˆ 4π0 r for r < R ~ Esphere = 0

11 / 30 2 A uniformly charged sphere at locations outside the sphere produces the same E~ ﬁeld of a point charge with charge equal to the total charge on the sphere.

The charged sphere responds to applied electric ﬁelds in the same way as a point charge at its center would

Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere E~ from a Uniformly Charged Sphere

1 ~ 1 Q Note: for r > R, E = 2 rˆ. Does this formula look sphere 4π0 r familiar?

12 / 30 Deﬁnition Superposition Clicker Question Electric Field From a Dipole E~ from a Uniformly Charged Sphere E~ from a Uniformly Charged Sphere

1 ~ 1 Q Note: for r > R, E = 2 rˆ. Does this formula look sphere 4π0 r familiar?

2 A uniformly charged sphere at locations outside the sphere produces the same E~ ﬁeld of a point charge with charge equal to the total charge on the sphere.

The charged sphere responds to applied electric ﬁelds in the same way as a point charge at its center would

12 / 30 Deﬁnition: Dipole Moment Vector p~ p~ = q~s where q is the magnitude of both charges that make up a dipole, ~s is the position of the positive charge relative to the negative charge.

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary E~ ﬁeld of a dipole and the electric dipole moment vector

13 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary E~ ﬁeld of a dipole and the electric dipole moment vector

Deﬁnition: Dipole Moment Vector p~ p~ = q~s where q is the magnitude of both charges that make up a dipole, ~s is the position of the positive charge relative to the negative charge.

13 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary E~ ﬁeld of a dipole at other locations

Part of one of your labs will recreate this plot.

14 / 30 3 There are two lines of symmetry for a dipole, and we are going to derive simpler formulas along each of those lines.

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

1 Now we are going to apply the principal of superposition to get the electric ﬁeld of a dipole.

2 As a general hint, symmetry is your friend. Doing math is usually simpler when you somehow preserve or take advantage of a physical symmetry.

15 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

1 Now we are going to apply the principal of superposition to get the electric ﬁeld of a dipole.

2 As a general hint, symmetry is your friend. Doing math is usually simpler when you somehow preserve or take advantage of a physical symmetry.

3 There are two lines of symmetry for a dipole, and we are going to derive simpler formulas along each of those lines.

15 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

16 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

17 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

18 / 30 1 While the electric field of a point charge is proportional to r 2 , electric fields created by several charges may have a different distance dependence.

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

19 / 30 1 While the electric field of a point charge is proportional to r 2 , electric fields created by several charges may have a different distance dependence.

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

19 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Parallel to the Axis

1 While the electric field of a point charge is proportional to r 2 , electric fields created by several charges may have a different distance dependence. 19 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

20 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

21 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

22 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

23 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

24 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

25 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

26 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Perpendicular to the Axis

27 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Summary Dipole E~ Field

E~ Far From the Dipole

~ 1 p ~ 1 2p |Eperp| = 3 |Eonaxis | = 3 4π0 y 4π0 r

~ ~ For the same distance r, |Eonaxis | = 2 ∗ |Eperp|. At other locations you can still use superposition, it is just that the formula doesn’t simplify as nicely.

28 / 30 The force on the positive is equal in magnitude but opposite in direction of the force on the negative, so F~net = F~+ + F~− = ~0 So what could a dipole be used to measure? A dipole would experience a force in a non-uniform E~ ﬁeld. So they can measure the uniformity of the E~ ﬁeld

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Group Question 1

Dipole in a uniform E~ ﬁeld, What is the net force?

29 / 30 So what could a dipole be used to measure? A dipole would experience a force in a non-uniform E~ ﬁeld. So they can measure the uniformity of the E~ ﬁeld

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Group Question 1

Dipole in a uniform E~ ﬁeld, What is the net force?

The force on the positive is equal in magnitude but opposite in direction of the force on the negative, so F~net = F~+ + F~− = ~0

29 / 30 A dipole would experience a force in a non-uniform E~ ﬁeld. So they can measure the uniformity of the E~ ﬁeld

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Group Question 1

Dipole in a uniform E~ ﬁeld, What is the net force?

The force on the positive is equal in magnitude but opposite in direction of the force on the negative, so F~net = F~+ + F~− = ~0 So what could a dipole be used to measure?

29 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Group Question 1

Dipole in a uniform E~ ﬁeld, What is the net force?

The force on the positive is equal in magnitude but opposite in direction of the force on the negative, so F~net = F~+ + F~− = ~0 So what could a dipole be used to measure? A dipole would experience a force in a non-uniform E~ ﬁeld. So they can measure the uniformity of the E~ ﬁeld 29 / 30 The direction of E~ (i.e. Eˆ)

Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Group Question 2

Dipole in a uniform E~ ﬁeld

The E~ ﬁeld results in a torque on the dipole, aligning p~ to E~. Given this, what additional piece of information might a dipole measure?

30 / 30 Electric Dipoles Superposition Parallel to the Axis Electric Field From a Dipole Perpendicular to the Axis Dipole Summary Group Question 2

Dipole in a uniform E~ ﬁeld

The E~ ﬁeld results in a torque on the dipole, aligning p~ to E~. Given this, what additional piece of information might a dipole measure? The direction of E~ (i.e. Eˆ)

30 / 30