Lesson 5 Representing Fields Geometrically

Class 13 Today, we will: review characteristics of field lines and contours learn more about electric field lines and contours of point charges learn more about magnetic fields and contours of long, straight wires

Section 1 Review of Field Lines from Lesson 2

Review from Lesson 2 For point charges: –Electric fields are obtained from aligning threads –Magnetic fields are obtained from aligning stubs.

Review from Lesson 2 For point charges: –Electric fields are radially outward from a + source and inward toward a – source. –Magnetic fields form circles around the path of the source. The direction is given by a right-hand rule.

Field Lines and Threads Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ. A ℓ

A Field Lines and Threads Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ. Each thread is part of one field line when aligned. ℓ

A Field Lines and Threads Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ. Each thread is part of one field line when aligned. The force is proportional to the density of threads time the length of the threads. ℓ

A Field Lines and Threads Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ. Each thread is part of one field line when aligned. The force is proportional to the number of field lines per unit area. ℓ

Field Lines To find the strength of a field from the field lines: 1)Take a small section of a contour surface of area A. 2)Count the number of field lines passing through: N. 3) The field is proportional to N/A. A

Field Contours Elements of the field contour are surfaces that 1) are perpendicular to the field lines. 2) are spaced so that the field is stronger where the surfaces are close together.

Field Contours To find the strength of a field from the field contour: 1)Take a small section of a field line of length L. 2)Count the number of contour surfaces pierced by the line segment: N. 3) The field is proportional to N/L.

Section 2 The Electric Field of Point Charges

Electric Field The fundamental source of electric field is a point charge. All electric fields arise from a collection of point charges. We will only consider static electric fields for now.

Electric Field of a Point Charge Radially outward for positive charges. Radially inward for negative charges. Uniform in all directions in space. Number of field lines is proportional to the magnitude of the charge.

Electric Field of a Point Charge

Electric Field of a Point Charge “Flat” Representation

Coulomb’s Law = Electric Field Lines Let’s use SI Units. If q =1C and r =1m and there are 4 field lines per coulomb: …but only when we draw 4 lines per C

Coulomb’s Law = Electric Field Lines If q = 1 there are 4 field lines. At If q = 3 there are 12 field lines. At →

Coulomb’s Law = Electric Field Lines There are twelve field lines. At →

Coulomb’s Law = Electric Field Lines “Coulomb’s Law is built into the geometry of the electric field lines.” All we had to do is choose the right number of lines, let them point radially outward or inward, and let them be uniformly distributed.

Electric Field Contours Electrical field contours are sets of spheres. We have to compute the separation distances between surfaces and place them in the right places “by hand.” “Electric field contours are unconstrained by geometry.”

Section 3 The Magnetic Field of Long, Straight Wires

Magnetic Field The fundamental source of magnetic field is an infinitely long, very thin, current-carrying wire. Not all magnetic fields arise from such wires, but their fields are simple enough we’ll use them anyway.

Magnetic Field of a Long Wire We calculated the field of a wire using stubs, but for now we’ll consider this an experimental result independent of Coulomb’s Law. “permeability of free space”

Magnetic Field Lines of a Long Wire Magnetic field lines form circles around a wire. The direction is given by the right-hand rule.

Magnetic Field Lines of a Long Wire Put your thumb in the direction of the current and the magnetic field lines will go around the wire in the direction of your fingers.

Magnetic Field Lines Magnetic field lines are sets of circles. We have to compute the separation distances between lines and place them in the right places “by hand.” “Magnetic field lines are unconstrained by geometry.”

Magnetic Field Contours of a Long Wire Magnetic field contours are sets of half-planes having one edge on the wire.

Magnetic Field in Two Dimensions We choose to draw three surfaces per ampere. We attach direction arrows to the surfaces in the direction of the field.

Wire Law = Magnetic Field Contours Let’s use SI Units. If i =1A and r =1m and there are 3 surfaces per ampere: …but only when we draw 3 lines per A

If i = 1 there are 3 field surfaces. At If i = 2 there are 6 field surfaces. At → Wire Law = Magnetic Field Contours

There are six field surfaces. At → Wire Law = Magnetic Field Contours

“The Wire Law is built into the geometry of the magnetic field contours.” All we had to do is choose the right number of surfaces, orient them correctly, and let them be uniformly distributed. Wire Law = Magnetic Field Contours

Class 14 Today, we will: learn how to draw the total electric field of two point charges. find that the electric field is like a single point in the near field and the far field. use symmetry to find the electric field lines of charged spheres, cylinders, and palnes.

Section 4 The Electric Field of Multiple Charges

Electric Fields - Threads A + charge and a ̶ charge each emit threads.

Electric Fields - Threads The threads of both charges combine to give a total force on a field particle.

Electric Field Lines Electric field lines point in the direction of the net force.

Electric Field Lines Take one vector pointing in the direction of the electric field near a positive charge.

Electric Field Lines Then attach the head of a second vector to the tail of the first vector.

Electric Field Lines Continue this process over and over.

Electric Field Lines This forms an electric field line for the pair of charges.

Electric Field Lines The direction of the electric field is tangent to the electric field line as before.

Electric Field Lines Recall that the electric field is stronger where the lines are closer together. stronger field weaker field

How to Draw Electric Field Lines of Multiple Charges

Two Point Charges Let’s take charges of +2 C and -2 C.

Near Field What will the field be like near the positive charge?

Hint: Don’t draw field lines that go directly toward or away from another charge. Near Field

What will the field be like near the negative charge? Near Field

Far Field What will the field be like far from the two charges?

Far Field There is no field (or at least it’s very weak).

Two Point Charges Charges of +2 C and -2 C.

Two Point Charges Charges of +2 C and -2 C.

Two Point Charges Charges of +2 C and -2 C.

Two Point Charges Charges of +2 C and -2 C.

Two Point Charges Charges of +2 C and -2 C.

Two Point Charges Charges of +2 C and -2 C.

Two Point Charges Let’s go to ActivPhysics – so we can adjust parameters easily. Remember to consider the near field, the far field, and then the intermediate field.

Two Point Charges

Clicker Question 1 Is this picture OK? A – yes B -- no +1 –1

Clicker Question 1 Is this picture OK? A – yes B -- no +1 –1 Field lines can not cross. The near field is incorrect.

Clicker Question 2 Is this picture OK? A – yes B -- no +2 –1

Clicker Question 2 Is this picture OK? A – yes B -- no +2 –1 The far field is incorrect. The near field isn’t drawn well.

Section 5 Electric Field Lines and Symmetry

Symmetry What can we say about the electric field lines of a sphere?

Spherically Symmetric Charge How do the electric field lines of a spherically symmetric charge compare to a point charge? If the charge is the same, the total number of field lines is the same.

Spherically Symmetric Charge Outside of the sphere:

Cylindrically Symmetric Charge What do the field lines of a cylindrically symmetric charge look like?

Cylindrically Symmetric Charge What do the field lines of a cylindrically symmetric charge look like?

Cylindrically Symmetric Charge How do the electric field lines of a cylindrically symmetric charge compare to a point charge? If the charge in length L is the same as the point charge, the total number of field lines coming from length L is the same as from the point charge.

Cylindrically Symmetric Charge point charge with a charge q. infinitely long rod with a charge q on a segment of length L

Cylindrically Symmetric Charge ←same N → E outside cylinder→ Draw a sphere around the point charge

Cylindrically Symmetric Charge ←same N → E outside cylinder→ Draw a cylindrical can of length L around the cylinder

Planar Symmetric Charge What do the field lines of an infinite plane of charge look like?

Planar Symmetric Charge What do the field lines of an infinite plane of charge look like?

Planar Symmetric Charge How do the electric field lines of a infinite plane of charge compare to a point charge? If the charge in area A is the same as the point charge, the total number of field lines coming from area A is the same as from the point charge.

Cylindrically Symmetric Charge point charge with a charge q. infinite plane with a charge q on a segment of area A

Planar Symmetric Charge ←same N → Use a spherical surface again but N/2 lines on each side of the plane!

Planar Symmetric Charge ←same N → but N/2 lines on each side of the plane!

Threads (point) vs. Field Lines (bulk) very small, move radially outward from charges start on positive charges and end on negative charges – or go off to infinity move at the speed of lightchanges “move” at the speed of light go in straight linesbend around, following the direction of the force can not be destroyed, but their effects can be cancelled by other threads do not exist in regions where there is no net force magnetic fields result from motion correction to threads magnetic fields result from 1)currents (unrelated to electric fields) or from 2)changing electric fields – but we won’t see the second case until later.

Class 15 Today, we will: learn to visualize magnetic field lines for two wires. learn to visualize electric and magnetic field contours for two charges or wires.

Symmetry and Field Lines 1) If we know how much charge an object has, we know how many field lines there are. 2) If there is enough symmetry (spheres, cylinders, planes), we know what the field lines look like. 3) If we know where all the field lines are, we can measure the number of field lines per unit area anywhere in space, so we know the electric field everywhere!

Section 6 Magnetic Field Lines and Electric Field Contours of Multiple Sources

What about Magnetic Field Contours? Their pictures are not exactly the same as electric field lines, but if we want close instead of exact, we can draw them the same way.

What about Magnetic Field Contours? Same charges (left) and same currents (right). electric field contours electric field lines magnetic field contours magnetic field lines

What about Magnetic Field Contours? Opposite charges (left) and opposite currents (right). electric field contours electric field lines magnetic field contours magnetic field lines

Magnetic Field Contours Conclusion: If you can draw electric field lines for two charges, you can draw magnetic field contours for two currents! electric field contours electric field lines magnetic field contours magnetic field lines

What about Electric Field Contours and Magnetic Field Lines? electric field contours electric field lines magnetic field contours magnetic field lines

How do we make electric field contours? Start with the electric field lines.

How do we make electric field contours? Add spherical contours in the near field.

How do we make electric field contours? The next surfaces are deformed – closer where the field is stronger.

How do we make electric field contours? Then a single surface passes around both charges.

How do we make electric field contours? The surfaces become more spherical…

How do we make electric field contours? until they become spheres in the far field.

Now take the case of opposite charges. Start with the electric field lines.

Now take the case of opposite charges. Add spherical contours in the near field.

Now take the case of opposite charges. The next surfaces are deformed – closer where the field is stronger.

Now take the case of opposite charges. The next surfaces are deformed – closer where the field is stronger.

Now take the case of opposite charges. There are no surfaces nor lines at infinity in this case.

Diagrams for magnetic field look similar to those of the electric field. The field lines and contours reverse roles, however!

Diagrams for magnetic field look similar to those of the electric field. Two equal currents going in the same direction. field lines field contour

Diagrams for magnetic field look similar to those of the electric field. Two equal currents going in opposite directions. field lines field contour