Studio Session 1

Vectors and vector fields

In physics, quantitative models are developed on the basis of measurements.  Measurements are made in standard increments, called units.  Without units, a measurement is meaningless.  Many quantities are specified by a magnitude (a number and the appropriate unit) and a direction in space.  Such quantities are called vector quantities.  Symbols that denote these vector quantities are bold letters, or normal letters with arrows drawn above.

Examples of vector quantities:

displacement (d):   d = 10 m north
velocity (v): v = 3 m/s eastward
acceleration (a) a = 6 m/s2 west
force (F) F = 9 N up

To uniquely specify vector quantities we need a reference point and reference lines, i.e.  we need a coordinate system.  The most commonly used coordinate systems are rectangular, Cartesian coordinate systems.  Other widely used coordinate systems are cylindrical and spherical coordinate systems.

This semester we will study electromagnetic interactions.  To find the electric and magnetic fields produced by charged particles and the electric and magnetic forces acting on objects, we have to perform vector operations.  In this studio session we will review these vector operations and discuss ways to visualize vector fields.

Link:  Vectors - Fundamentals and Operations

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  This log will become your lab report.  After each step (a) - (k), write down what you have observed.  Address the points highlighted in blue.  Answer all questions.


Representing vectors

In Cartesian coordinates a vector is represented by its components along the axes of the coordinate system.
Example:  F = (Fx, Fy) = Fxi + Fyj  = 3 N i - 4 N j.
Here i and j are unit vectors.  Unit vectors have magnitude 1 and no units.  They are used as direction indicators.
(i, j, k point in the x-, y-, and z-direction, respectively.)

In the polar coordinates, in two dimensions, a vector is represented by its magnitude and the angle its direction makes with the x-axis.
Example:  F = (F, φ) = (5 N, 306.87o) = (5 N, -53.13o)
Cylindrical coordinates and spherical coordinates are two different extensions of polar coordinates to three dimensions.

Exercise 1:  Converting between representations.

Your instructors will ask you to solve several conversion problems.  Discuss your answers with your neighbors and then enter them into your log.


Adding and subtraction vectors

To add or subtract physical vectors, they have to have the same units.  To find the sum of two physical vector quantities with the same units algebraically, we add the x, y, and z-components of the individual vectors.

Example:
Let vector A have components (3, 4) and vector B have components  (2, -3).  Let C = A + B be the sum of the two vectors.  Then the components of C are (3+2, 4+(-3) = (5, 1). 
The magnitude of the vector C is C = (25 + 1) = 5.1, and the angle C makes with the x-axis is φ = tan-1(1/5) = 0.197 rad = 11.3o.
To subtract vector B from vector A we subtract the components of vector B from the components of vector A.

Exercise 2:  Vector addition and subtraction.

Your instructors will ask you to solve several vector addition and subtraction problems.  Discuss your answers with your neighbors and then enter them into your log.


Multiplying vectors

Vectors can be multiplied by a scalar (or number).  Multiplying  a vector by a scalar changes the magnitude of the vector, but leaves its direction unchanged.

Example: 
F = (3 N, -4 N),  3F =  3*(3 N, -4 N) = (9 N, -12 N),
or
F = (F, φ) = (5 N, 323.13o),  3F = (15 N, 323.13o).

A vector can also be multiplied by another vector.  There are two different products of vectors.

The scalar product or dot product of two vectors A and B is a scalar quantity (a number with units) equal to the product of the magnitudes of the two vectors and the cosine of the smallest angle between them.
A∙B = ABcosθ.
In terms of the Cartesian components of the vectors A and B the scalar product is written as
A∙B = AxBx + AyBy + AzBz.
In one dimension, the scalar product is positive if the two vectors are parallel to each other, and it is negative if the two vectors are anti-parallel to each other, i.e. if they point in opposite directions.

cross product illustrationThe vector product or cross product of two vectors A and B is defined as the vector C = A B
The magnitude of C is C = AB sinθ, where θ is the smallest angle between the directions of the vectors A and B

C is perpendicular to both A and B, i.e. it is perpendicular to the plane that contains both A and B.
The direction of C can be found by using the right-hand rule.
Let the fingers of your right hand point in the direction of A.
Orient the palm of your hand so that, as you curl your fingers, you can sweep them over to point in the direction of B.
Your thumb points in the direction of C = A B.

right-hand ruleIf A and B are parallel or anti-parallel to each other, then C = A B = 0, since sinθ = 0.
If A and B are perpendicular to each other, then sinθ = 1 and C has its maximum possible magnitude.
We can find the Cartesian components of C = A B in terms of the components of A and B.

Cx = AyBz - AzBy
Cy = AzBx - AxBz
Cz = AxBy - AyB

Exercise 3:  Vector multiplication

Your instructors will ask you to solve several vector multiplication problems.  Discuss your answers with your neighbors and then enter them into your log.


Vector fields

The gravitational field
Forces are vectors.  A force that we are familiar with is gravity.  Newton's law of gravity states that any two objects with mass m1 and m2, respectively, attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance r12 between them.

F
12 = (-G m1m2/r122) (r12/r12).

arrows showing two masses attracting each otherHere r12 = |r2 - r1| is the distance between mass m1 and m2, and (r12/r12) = (r2 - r1)/|r2 - r1| is the unit vector pointing from mass m1 to m2.
G is the gravitational constant, G = 6.67*10-11 Nm2/kg2.
The force F21, which mass m2 exerts on mass m1, is equal to -F12, according to Newton's third law.  The gravitational force is always attractive.

The point in an extended object from which the distance r12  is measured is its center of mass.  Mass m1 pulls on mass m2, and mass m2 pulls on mass m1.  The center of mass of each object is pulled towards the center of mass of the other object.

The gravitational force between masses decreases proportional to the inverse square of the distance between the masses.
Why do we not notice that inverse square dependence near the surface of Earth?

The radius of Earth is R = 6368 km.  If you climb a 1000 m high mountain, your distance from the center of the earth changes by (1/6368)*100 % = 0.016 % and the magnitude of the gravitational force acting on you changes by (1/6368)2*100 % = 2.4*10-6 %.  For all objects near the surface of Earth the distance from the center is nearly constant, and the magnitude of the gravitational force vector is therefore approximately constant and equal to GMearthmobject/R2 = mobjectg, with g = GMearth/R2 = 9.8 m/s2.  Over small distances, when the curvature of the earth's surface can be neglected, the direction of the gravitational force vector is also nearly constant.  It points straight downward towards the center of the earth.  The force of gravity acting on an object is called its weight. 

The gravitational force is not a contact force.  It acts at a distance.  We introduce the concept of the gravitational field to explain this action at a distance.  Massive particles attract each other.  We say that massive particles produce gravitational fields and are acted on by gravitational fields.  The magnitude of the gravitational field produced by a massive object at a point P is the gravitational force per unit mass it exerts on another massive object located at that point.  The direction of the gravitational field is the direction of that force.  The gravitational field produced by a point mass always points towards the point mass and decreases proportional to the inverse square of the distance from the point mass.  Near the surface of Earth the gravitational field produced by Earth is nearly constant and has magnitude F/m = g = 9.8 m/s2.  Its direction is downward.

To find the total gravitational field at a point calculate the vector sum of the gravitational fields produced by all masses that do not produce negligibly small gravitational fields at that point.


The electric field
All charged particles interact via the Coulomb force.  A particle with charge q1 exerts a force F12 on a particle with charge q2Coulomb's law gives this force as

F
12 = (keq1q2/r122) (r12/r12).

The constant ke is ke = 9*109 Nm2/C2.

arrows showing two equal charges repellin each otherThe force F21, which the particle with charge q2 exerts on the particle with charge q1, is equal to -F12, according to Newton's third law.
Charges can be positive or negative.  Two positively charged particles repel each other.  Two negatively charged particles repel each other.  But a positively charged particle and a negatively charged particle attract each other.

The Coulomb force is not a contact force.  It acts at a distance.  We introduce the concept of the electric field to explain this action at a distance.  We say that charged particle produce electric fields and are acted on by electric fields.  The magnitude of the electric field E produced by a charged particle at a point P is the electric force per unit positive charge it exerts on another charged particle located at that point.  The direction of the electric field is the direction of that force on a positive charge.  The actual force on a particle with charge q is given by F = qE.  It points in the opposite direction of the electric field E for a negative charge.
The electric field produce by a positive point charge always points away from the point charge and the electric field produce by a negative point charge always points towards the point charge.  The electric field decreases proportional to the inverse square of the distance from the point charge.
To find the total electrical field at a point calculate the vector sum of the electric fields produced by all charges that do not produce negligibly small electric fields at that point.


Velocity fields

The velocity of a fluid, for example the velocity of water flowing through a pipe or down a drain, is a vector field.  The velocity field describes the motion of a fluid at every point.  The length of the flow velocity vector at any point is the flow speed.


Graphical representations of vector fields

One way to graphically represent a vector field in two dimensions is by drawing arrows an a grid.  Set up a grid and find the magnitude and direction of the field vector at every grid point.  At each grid point draw an arrow with the tail anchored at the grid point and a length proportional to the magnitude of the vector in the direction of the field vector.

Examples:

Velocity field of an ideal fluid in a pipeVelocity field of an ideal fluid in a pipe
Continuity equation: A1v1 = A2v2
A2 = A1  -->  v2 = 2v1
The arrows in the narrower section of the pipe
are twice as long as the arrows in the wider section.

Gravitational field near the surface of EarthGravitational field near the surface of Earth
g = 9.8 m/s2 = constant, pointing downward.
All arrows have the same length.

Electric field of a positive point charge at the originElectric field of a positive point charge at the origin
Note how fast the field decreases as a function of the distance
from the point charge as a consequence of the 1/r2 dependence.
Arrows near the origin are not drawn, because they are too long.
The magnitude of the field approaches infinity as we approach
the origin.

 

Exercise 4:  Drawing arrows to represent field

Electric field applet: http://physics.weber.edu/schroeder/software/EField/
Note:  If Java is blocked, add http://physics.weber.edu to the Exception Site List in the Java Control Panel.
The Edge browser does not support Java.  In Windows 10 set another browser as your default browser.

Use this applet to produce arrow representations of the electric field of some charge distributions.

Explore the interface!
You can place charges onto the canvas.  With the hand you can move them, with the eraser square you can remove them. 
The arrow is used to display the magnitude and direction of the electric field at any point. 
Under patterns you can select field grid to display the field at grid points.

Clear all charges and arrows and the show the field grid.

Drawing arrows to represent field

Place a positive charge at position 1.  Describe the arrow representation of the electric field.

Place a second positive charge at position 2.  Describe the arrow representation of the electric field.

Move the second charge horizontally towards the first.  Describe the changes in the arrow representation.

For the relative position of the charges shown below, describe the field vectors whose tails lie on the red line.

two positive charges

Can you explain their magnitude and direction using vector addition of the field vectors of the two individual charges?

Replace the positive charge at position 2 with a negative charge.  Repeat!


The arrow representation for the field produced by more than one source can become quite messy.  Another way to graphically represent a vector field is by drawing field lines.  The direction of the field at any point is given by the direction of a line tangent to the field line, while the magnitude of the field is given qualitatively by the density of field lines.  Field lines can emerge from sources and end in sinks, or they can form closed loops. 
To draw a field line calculate the field at a point. 
Draw a short line segment (Δl --> 0) in the direction of the field. 
Recalculate the field at the end of the line segment. 
Repeat.

Examples:

Velocity field lines or streamlines for a liquid flowing in a pipeVelocity field lines or streamlines for a liquid flowing in a pipe.
The density is higher in region 2 where the velocity of the liquid has a greater magnitude.

Field lines of the gravitational field near the surface of Earth
Field lines of the gravitational field near the surface of Earth.  The lines are evenly spaced since the field is constant.

field lines for point charges
Electric field lines for a positive (source) and for a negative charge (sink).
The number of lines emerging from or converging at the charge is proportional to the magnitude of the charge.

Exercise 5:  Drawing field lines to represent field

Download (or obtain from your instructor) a copy of this word document containing diagrams of several charge configurations.  Draw field lines on the diagrams.  Field lines should leave or enter a charge symmetrically, and the number of lines entering or leaving should be proportional to the magnitude of the charge.  Choose a reasonable proportional factor.  You can use the applet below as a guide.  Choose to display only the electric field lines in the applet.

Electric field lines: http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html
Note:  If Java is blocked, add http://www.cco.caltech.edu to the Exception Site List in the Java Control Panel.

Answer the following questions.

If there is no field line drawn at a particular point, does this mean there is no field at that point?  Explain!

As you approach a field line, does the field get stronger?  Explain!

Is it possible for two field lines to cross each other?  Explain!


Convert your log into a session report, certify with you signature that you have actively participated, and hand it to your instructor.