MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics
8.02
Problem Solving Session 1: Electric Dipoles and Torque
Section
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Introduction: In the first problem you will learn to apply Coulomb’s Law to find the electric field of an electric dipole. You will investigate the properties of the electric field associated with an electric dipole. You will also explore what happens to an electric dipole when it is placed in an uniform electric field.
Readings: Course Notes 8.02: Chapter 2 Coulomb’s Law Section 2.7-2.8 Introduction Electric Dipole
An electric dipole consists of two equal but oppositely charged point-like objects, +q and −q, separated by a distance 2a, as shown in Figure 1.
The dipole moment vector p which points from −q to +q (in the +y-direction) is given by ˆ 2qa = p j (1)
The magnitude of the electric dipole is p=2qa, where q>0. For an overall charge-neutral system having N charged objects, the electric dipole vector p is defined as
p≡ qiri i=1 i=N
∑
(2)where ri is the position vector of the charged object with charge qi. Problem 1 Electric field of a Dipole
Question 1: Consider the electric dipole moment shown in Figure 1. Find the x- and y-components of the electric field at a point with coordinates (x,y,0), using
E+ =keq+ r+3 r+ and E− =keq− r−3 r−.
Solution: From the figure we have that the vectors from the charged particles to the field point are
r± =xˆi+(ya)ˆj
The respective distances from the positive and negatively charged objects to the field point at (x,y,0) are given by the expressions
r± =(x 2 +(y
a)2)1/ 2
The charges are q+ =q and q− =−q. Therefore the superposition of the two electric fields yields E=E++E− =keq+ r+3 r++keq− r−3 r− =keq xˆi+(y−a)ˆj (x2+(y−a)2)3/ 2 − xˆi+(y+a)ˆj (x2 +(y+a)2)3/ 2 ⎛ ⎝⎜ ⎞ ⎠⎟
Ex =keq x x2 +(y−a)2 ⎡⎣ ⎤⎦3/ 2 − x x2 +(y+a)2 ⎡⎣ ⎤⎦3/ 2 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟. Similarly, the y-component is given by
Ey =keq y−a x2 +(y−a)2 ⎡⎣ ⎤⎦3/ 2 − y+a x2 +(y+a)2 ⎡⎣ ⎤⎦3/ 2 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟
We can show that the electric field of the dipole in the limit where r>>a is Ex = ke3p
r3 sinθcosθ, Ey = kep
r3 (3cos
2θ −1) (3)
where sinθ = x/r and cosθ = y/r. See Chapter 2.14.4 Electric Field of a Dipole. Question 2: By what power of distance does the strength of electric field fall off? How does this compare to a single point charge? Briefly explain a reason for the difference between these two cases.
Solution: The electric field E due to a dipole decreases with r as 1/r3, unlike the 1 /r2 behavior for a point-like charged object. This is to be expected since the net charge of a dipole is zero and therefore must fall off more rapidly than 1 /r2 at large distance. The exact electric field lines due to the oppositely charged objects are shown in the figure on the left in Figure 2. The electric field lines corresponding to the electric dipole given by Eq. (3) are shown in the figure on the right in Figure 2.
Question 3 Electric Dipole Animation Open up the applet, Two Point Charges, http://peter-edx.99k.org/PCharges.html
The units for charge are µC . Create an electric dipole with charges q1 =3.0µC and q2 = −3.0µC. Click on the Electric Fields: Grass Seeds bar to see a representation of the electric field. Note that this applet can shows the three different representations we use for vector fields: (1) vector field grid of arrows, (2) field lines, and (3) grass seeds.
Problem 2 Electric Dipole in a Uniform Electric Field
Place an electric dipole in a uniform field E = Eˆi, with the dipole moment vector p making an angle θ with the x-axis.
Question 1: What is a vector expression for the dipole moment p? Give you answer in the form p = pxˆi+ py ˆj where you determine the components (px,py).
Solution:
From Figure 3, we see that the unit vector which points in the direction of p is
cosθˆi+sinθˆj. Thus, we have
ˆ ˆ
2 (cosqa θ sinθ )
= +
p i j (1)
Question 2: Find a vector expression for the torque on the dipole. What point did you compute the torque about? Is your answer independent of that choice of point?
Solution: As seen from Figure 3 above, since each point-like charged object experiences an equal but opposite force due to the field, the net force on the dipole is Fnet =F++F− =0. Even though the net force vanishes, the field exerts a torque a toque on the dipole that is independent of a choice of point. We select the midpoint O of the dipole about which to calculate the torque.
τ=r+ ×F++r− ×F− =(acosθˆi+asinθˆj)×(F+ˆi)+(−acosθˆi−asinθˆj)×(−F−ˆi)
=asinθF+(−kˆ)+asinθF
−(−kˆ) =2aFsinθ(−kˆ)
(2)
Question 3: What type of motion does the dipole undergo if it is released from its position and is free to move?
Solution: The direction of the torque is −kˆ, or into the page. The effect of the torque τ is to rotate the dipole clockwise so that the dipole moment p becomes aligned with the electric field E .
Question 4: Show that τ=p×E .
Solution: With F =qE, the magnitude of the torque can be rewritten as
2 (a qE)sin (2 ) sinaq E pEsin
τ = θ = θ = θ
and the general expression for torque becomes τ=p ×E . Question 5: Torque on an Electric Dipole Animation Open up the applet, Torque on an Electric Dipole
http://web.mit.edu/viz/EM/visualizations/electrostatics/ForcesTorquesOnDipoles/torqueondipolee/torqueondipolee.htm The units for the dipole moment are 10−6m⋅C and the units for electric field strength are
10−3N⋅C-1 Set the dipole moment p = 0.05×10−6m⋅C , the electric field strength
E = 0.1×10−3N⋅C-1, and the damping to zero. What approximately is the period of oscillation? Click on the Electric Fields: Grass Seeds bar to see a representation of the electric field at various moments in the oscillation. At what point in the cycle are the field lines most contorted? At what point are the field lines nearly straight (except close to the dipole).
Summary
• Two equal but opposite point-like charged objects form an electric field that far from the charged objects is an electric dipole. The electric dipole moment vector p points from the negative point-like charged object to the positive point-like charged object, and has a magnitude
2 p= aq
• The torque acting on an electric dipole p placed in a uniform electric field E is
