A Dipole with its scalar and vector fields A G Inc. Click within the blue frame to access the control panel. Only use the control panel (at the bottom of screen) to review what you have seen. This presentation is partially animated. When using your mouse, make sure you click only when it is within the blue frame that surrounds each slide.

- The location of the observer or “test” charge that is investigating the situation. + q q A dipole with equal but opposite charges separated by a short distance. An arrow that represents the dipole moment vector associated with this dipole. The magnitude (length) of this vector is product of the amount of charge multiplied by the distance between the split charges. The dipole is the entity that actually exerts the force that will repel (or perhaps attract) another charge entity. The dipole moment is the mathematical construct that was created to explain the preferential orientation the dipole has relative to the orientation of this “test” charge entity. You can use Coulomb’s law to calculate the force exerted at some location a distance away from the dipole itself. That force can also be related to the voltage (in volts) associated with the situation. To expedite the learning points of this presentation we will just assume that someone knows the scalar point function that will let us calculate those voltage values.

z + q - q The location of the observer or “test charge that is investigating the situation. In this example, only the plane defined by the unit vectors |j> and |k> is shown. y

z + q - q The location of the observer or “test charge that is investigating the situation. In this example, only the plane defined by the unit vectors |j> and |k> is shown. y |a r >

+ q - q z 1 |k> z r 1 |a r > |a r > < a r = 1 y 1 |k> y r 1 |a r > = ? y 1 |j> z 1 |k> +

+ q - q z The voltage value on the blue curve is constant V 7 |a r > z 1 |k> r 1 |a r >

+ q - q z The voltage value on the blue curve is constant The voltage value on the orange curve is constant but a negative of the blue curve value. V 7 V 7 |a r > The voltage value on either the blue curve or the orange curve is constant and determined by a scalar “point” function. z 1 |k> An example of an equipotential line (curve). The values produced by a scalar point function are always scalars, NEVER vectors. r 1 |a r >

+ q - q z V 7 V 7 |a r > z 1 |k> An example of a set of equipotential lines (curves). V 6 V 7 V 6 does not equal r 1 |a r >

+ q - q z V 7 V 7 |a r > z 1 |k> An example of a set of equipotential lines (curves). V 6 V 7 V 6 does not equal The solid blue curves map the voltage values (scalars) of a voltage scalar point function onto the scalar field (the region around the dipole). The solid orange curves map the negative values of voltage onto the scalar field (the region around the dipole). An example of a scalar field for a scalar point function. r 1 |a r >

+ q - q z z 1 |k> Even though these curves are not orange, they still map the negative values of voltage onto the scalar field (the region around the dipole). Positive values of voltage on these equipotential lines. Negative values of voltage on these equipotential lines. - + r 1 |a r >

+ q - q z z 1 |k> 3 short vectors with each perpendicular to a point on the corresponding blue curve r 1 |a r >

+ q - q z z 1 |k> Red dots (dashes if you like) in each of these two curve indicate the collection of vectors that extend normal to each of the possible equipotential curves. r 1 |a r >

+ q - q z z 1 |k> 2 additional short vectors with each perpendicular to a point on the corresponding blue curve r 1 |a r >

+ q - q z z 1 |k> r 1 |a r >

+ q - q z z 1 |k> However, the dotted (dashed) red curves do map the position of the “tails” of the vector “arrows” in the vector field. An example of a vector field for a vector point function. No vectors are shown in this diagram. You have to see the vectors in your minds eye. (The dotted (dashed) lines are called stream lines.) The function that produces the vectors not shown in this diagram is called a vector point function. r 1 |a r >

+ q - q z z 1 |k> An example of a scalar field for a scalar point function. The function that produces the scalar values represented in each of the blue curves shown in this diagram is called a scalar point function. r 1 |a r >

+ q - q the “z” direction A dipole with no equipotential or stream lines shown. Both the scalar field and the vector field are “present” you just can not see them. the “r” direction

+ q - q Vector field for a vector point function. A dipole with the stream lines shown. the “z” direction the “r” direction

+ q - q the “z” direction the “r” direction Scalar field for a scalar point function. A dipole with the equipotential lines shown.

+ q - q The solid blue curves map the values of voltage in the scalar field (the region around the dipole). The dotted red curves map the position of the “tail” of the vector “arrows” in the vector field (the region around the dipole moment). the “z” direction the “r” direction A dipole with both equipotential and stream lines shown.

+ q - q The function that produced the values for the solid blue curves is a scalar point function. the “z” direction the “r” direction Four things to remember to keep scalar and vector fields clear in your mind The function that produced the vectors that are not shown in the dotted red curves is a vector point function. The dotted red curves do not show vectors but do indicate where the “tails” of the vectors are. (If you like, each dash is the tail of a vector.) (1) (3)(2) (4) In a x, y, and z situation, the blue curves become equipotential surfaces. The values produced by a scalar point function are always scalars, NEVER vectors.

Another Example – The induced dipole An electric field with an electric field intensity < E o | = < E o |E o > 1/2 < k| = 1Note: Click your mouse to place an uncharged conducting sphere into this uniform electric field. Click your mouse to place a uniform electric field into the example. < k| z (Notice that this electric field is only in the “z” direction.) An electric field intensity streamline. (The field lines are solid red lines this time)

Sketch out the equipotential curves. < k| z < E o | = < E o |E o > 1/2 < k| = 1Note: Before proceeding; Sketch out the steam lines. Another Example – The induced dipole

< k| z The field has caused a separation of charge on the sphere Please notice that; The electric field lines (the stream lines) are always normal to any equipotential curve. The electric field lines have become distorted to accommodate the separation of charge. The magnitude of the electric field intensity vector inside the sphere is zero. Another Example – The induced dipole

< k| z An electric field intensity streamline. Please notice that; The electric field lines (the stream lines) are always normal to any equipotential curve. The electric field lines have become distorted to accommodate the separation of charge. The magnitude of the electric field intensity vector inside the sphere is zero. The field has caused a separation of charge in the sphere A voltage equipotential line. What will happen when you pull the sphere out of the electric field? Another Example – The induced dipole

An electric field with an electric field intensity < E o | = < Eo|Eo > 1/2 < a z | The electric field lines become uniform again. Please notice that; The charge separation on the sphere disappears. The equipotential curves also disappear. (They are not there which is different from not seeing them.) Another Example – The induced dipole

End of Presentation A G Inc.