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Details on dipoles

Page history last edited by Joe Redish 1 year, 9 months ago

8.2.P13

 

Prerequisites:

 

Since positive and negative charges attract each other strongly, in neutral matter their effect tend to cancel out producing no external electric effects. But when the positive and negative charges become slightly separated for whatever reason, the resulting structures can have important electric effects.

 

For example, many molecules, despite the fact that they are electrically neutral overall, can have one side that's electrically negative and one side positive (a water molecule, for example). We call such things "electric dipoles," and we can model them as pairs of particles of charge + q and -q (where q is a positive number) separated by a distance d. Usually, d is a very small distance. (For water it would be around 10-11 m, thinking of a few protons worth of charge on one end - about 6 x 10-19 C -- and a few electrons worth at the other.) Furthermore, because of the magic of quantum mechanics, in many molecules it behaves more like a rigid rod than like a soft spring. So we can treat d as a fixed distance.*

Electric dipoles play critical roles in the electrical properties of ionic fluids (such as inside biological systems). They also provide useful models for analyzing the electrical response of the human heart and in understanding electrocardiograms.

 

In this problem, we'll explore the electrical properties of a dipole using a variety of representations: words, diagrams, and equations.

 

Suppose you have a dipole that's free to move in any way (including rotate - imagine it floating in space). And there's an object with charge Q a distance r away. That distance r would be much larger than d, the distance between the charges of the dipole, so we draw the dipole small.

 

a) Consider the forces between the charge Q and the dipole. If the dipole is free to rotate:

  • Would the dipole end up attracted to the charge, repelled, or neither?
  • Would the charge Q be attracted, repelled, or neither to the dipole?
  • Do your answers depend on the sign of Q?

 

Now let's explore the effect that the dipole has on charges in the region around it by studying the electric field and potential due to a dipole.

 

Bring up the PhET simulation, Charges and Fields.Click on the "hamburger" (stacked three-lines) at the lower right of the screen. Select "Options" from the pop-up menu and click on the box "Projector mode". (This takes you out of black-screen mode and will save a ton of ink when you print images.) Turn on the "Grid" in the control box on the upper left and turn off "electric field". 

 

We'll take a point in the center of the grid as the origin of our coordinate system and put a dipole centered at the origin. Place a +1 nc (red) charge along the x axis at the position 0.5 (½ box to the right) and a -1 nc (blue) charge along the x axis at the position -0.5 (½ box to the left). Your screen should look like the figure below.

 

 

 

(The axes do not appear on the program but are shown here for reference.)

 

b) Grab a sensor (small yellow dot) from the charge palette at the bottom of the grid and explore what the direction of the electric field you would measure along the x-axis and along the y-axis. Explain what direction electric force a positive and a negative ion would feel if placed at various places along these axes.

 

c) Now that you have a sense of what forces a dipole exerts on both positive and negative charges, can you explain why both positive and negative charged objects attracts neutral matter?

 

d) Turn on the electric field by checking the box in the palette at the upper right. The strength of the field is displayed in this plot by the intensity of the color of the vector so you can't really tell how it goes quantitatively. Include a screenshot of the dipole field in your assignment and make a plot of the strength of field along the x and y axes from 1.5 to 3.5 in steps of 0.5. Use the sensor and measure the length of the arrow as a way to determine this.

 

e) Now turn off the electric field, put the sensor back in the palette and measure the potential by grabbing the voltmeter from the measurement tools palette on the right. Plot the electric potential along the x and y axes from -3.5 to +3.5 in steps of 0.5. (On the x axis, skip the points where there are charges.)

 

f) Another way to explore the electric effects of the dipole is to make a contour plot of the electrostatic potential. To do this, move the crosshairs of the voltmeter until the voltage reads a nice number. Then click the pencil on the voltmeter. This shows all points where the potential has that value -- an equipotential. Clicking on a point at +10 V gives the diagram at the right.

Now add equipotentials for a uses sequence of values, such as (+15, +10, +5, +4, +3, +2, +1, 0, -1, -2, -3, -4  -5, -10, -15, for example). (These loops are really surfaces since in 3D you can rotate the whole picture about the x axis) Include a screen shot of your diagram in your assignment.

 

g) The equipotential diagram you have created in part (f) is like a topographical map describing hills and valleys, with the potential being analogous to the height. Describe your map in terms of hills and valleys and use it to describe how a positive ion would start to move if placed at various positions on your diagram.

 

h) The equipotential map is not exactly equivalent to a topographical map. Describe how a negative ion would start to move if placed at various positions on your diagram.

 

i) Given that the electrostatic potential energy between two point charges is given by U = QV = kCqQ/r, suppose the dipole is made up of charges +q and -q (where "q" is positive) separated by a distance d. Write an equation for the electrostatic energy of a point charge, Q, placed  at coordinates (x,y) expressed in terms of kC, q, Q, d, x, and y.

 

j) Sketch a graph of the potential energy U of the charge Q as it moves up and down the y axis. Explain why it looks as it does.

 

 

Joe Redish 2/14/20

 

 

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