### 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/s^{2} 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** = (F_{x}, F_{y}) = F_{x}**i**
+ F_{y}**j** = 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, 306.87^{o}) = (5 N,
-53.13^{o})

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)^{½} = 5.1,
and the angle **C** makes with the x-axis is φ = tan^{-1}(1/5) =
0.197 rad = 11.3^{o}.

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), 3**F** = 3*(3 N, -4 N) = (9 N, -12 N),

or

**F** = (F, φ) = (5 N, 323.13^{o}), 3**F** = (15 N, 323.13^{o}).

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 **= A_{x}B_{x }+ A_{y}B_{y
}+ A_{z}B_{z}.

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.

The **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**.

If **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.

C_{x }= A_{y}B_{z }- A_{z}B_{y}

C_{y }= A_{z}B_{x }- A_{x}B_{z}

C_{z }= A_{x}B_{y }- A_{y}B_{x }

**
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 m_{1} and m_{2}, respectively, attract each
other with a force proportional to the product of their masses and inversely proportional
to the square of the distance r_{12} between them.

F_{12} = (-G m_{1}m_{2}/r_{12}^{2})
(**r**_{12}/r_{12}).

Here r_{12} =_{ }|**r**_{2} - **r**_{1}|
is the distance between mass m_{1} and m_{2}, and
(**r**_{12}/r_{12}) = (**r**_{2}
- **r**_{1})/|**r**_{2} - **r**_{1}| is the
unit vector pointing from mass m_{1} to m_{2.}

G is the gravitational
constant, G = 6.67*10^{-11 }Nm^{2}/kg^{2}.

The
force **F**_{21}, which mass m_{2} exerts
on mass m_{1}, is equal to -**F**_{12},
according to **Newton's third law**. The gravitational force is always
attractive.

The
point in an extended object from which the distance r_{12} is measured is its center of mass.
Mass m_{1}
pulls on mass m_{2}, and mass m_{2} pulls on mass m_{1}.
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 GM_{earth}m_{object}/R^{2}
= m_{object}g, with g = GM_{earth}/R^{2}
= 9.8 m/s^{2}. 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/s^{2}. 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 q_{1
}exerts a force **F**_{12} on a particle with charge q_{2}.
**Coulomb's law** gives this force as

F_{12} = (k_{e}q_{1}q_{2}/r_{12}^{2})
(**r**_{12}/r_{12}).

The constant k_{e} is k_{e }= 9*10^{9 }Nm^{2}/C^{2}.

The force **F**_{21}, which the particle with charge q_{2}
exerts on the particle with charge q_{1}, is equal to -**F**_{12},
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**
= q**E**. 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 pipe**

Continuity equation: A_{1}v_{1} = A_{2}v_{2
}A_{2} = ½A_{1} --> v_{2} = 2v_{1
}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 Earth**

g = 9.8 m/s^{2} = constant, pointing downward.

All arrows have the same length.

**Electric 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/r^{2}
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.

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.

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 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. The lines are evenly spaced since
the field is constant.

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.