Electromagnetism: Electric Fields, Magnetic Fields, and Electromagnetic Induction, Study notes of Electromagnetism and Electromagnetic Fields Theory

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Electricity

Summary

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IV

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• ELECTRIC CHARGE AND ELECTRIC FIELD ELECTROSTATICS
• COULOMB’S LAW
• ELECTRIC FIELD
• FIELD LINES
• ELECTRIC DIPOLES
• GAUSS’ LAW
• ELECTRIC FLIX
• GAUSS’ LAW
• ELECTRIC POTENTIAL
• ELECTRIC POTENTIAL ENERGY AND POTENTIAL DIFFERENCE
  • ELECTRIC POTENTIAL ENERGY
  • ELECTRIC POTENTIAL
• RELATION BETWEEN ELECTRIC POTENTIAL AND ELECTRIC FIELD
  • GENERAL RELATION BETWEEN V AND E..................................................................................................
  • ELECTRIC POTENTIAL DUE TO POINT CHARGE
  • ELECTRIC POTENTIAL DUE TO SEVERAL CHARGES
• ELECTRIC DIPOLE POTENTIAL
• MAGNETISM MAGNETISM AND ELECTROMAGNETISM
• MAGNETS AND MAGNETIC FIELD
• ELECTRIC CURRENT PRODUCE MAGNETIC FIELD; RIGHT-HAND RULE
• FORCE ON AN ELECTRIC CURRENT IN A MAGNETIC FIELD; RIGHT-HAND RULE
  • ELECTRIC CURRENT IN A MAGNETIC FIELD
  • RIGHT-HAND RULE
  • DEFINITION OF MAGNETIC FIELD
• FORCE ON AN ELECTRIC CHARGE MOVING IN A MAGNETIC FIELD; RIGHT-HAND RULE
• RIGHT-HAND RULES (TABLE)
• LORENTZ EQUATION...........................................................................................................................
• MAGNETIC DIPOLE MOMENT
• THE HALL EFFECT...............................................................................................................................
• SOURCES OF MAGNETIC FIELD
• MAGNETIC FIELDS DUE TO ONE AND TWO WIRES
  • MAGNETIC FIELD DUE TO A STRAIGHT WIRE
  • FORCE BETWEEN TWO PARALLEL WIRES
• DEFINITION OF AMPÈRE AND COULOMB
• AMPÈRE’S LAW
• MAGNETIC FIELD OF A SOLENOID
• BOIT-SAVART LAW.............................................................................................................................
• ELECTROMAGNETIC INDUCTION
• INDUCED ELECTROMOTIVE FORCE (EMF)
• MAGNETIC FLUX
• FARADAYS’ LAW OF INDUCTION
• LENZ’S LAW
• TRANSFORMERS................................................................................................................................
• GENERAL FORM OF FARADAY’S LAW
• MAXWELL’S EQUATIONS AND ELECTROMAGNETIC WAVES (EMW)
• GENERAL FORM OF AMPÈRE’S LAW
• GAUSS’ LAW FOR MAGNETISM
• MAXWELL’S EQUATIONS
• PRODUCTION OF ELECTROMAGNETIC WAVES
• ENERGY IN ELECTROMAGNETIC WAVES; THE POYNTING VECTOR
  • ENERGY STORED IN AN ELECTROMAGNETIC WAVE
  • THE POYNTING VECTOR
• RADIATION PRESSURE
• CAPACITANCE, DIELECTRICS, ELECTRIC ENERGY STORAGE ELECTRIC CIRCUITS AND CURRENTS
• CAPACITOR
• CAPACITORS IN SERIES AND PARALLEL
• ELECTRIC ENERGY STORAGE
• DIELECTRICS
• ELECTRIC CURRENTS AND RESISTANCE
• ELECTRIC CURRENT
• OHM’S LAW: RESISTANCE AND RESISTIVITY
• RESISTIVITY......................................................................................................................................
• ELECTRIC POWER
• CURRENT DENSITY AND DRIFT VELOCITY..................................................................................................
• INDUCTANCE AND ENERGY OF MAGNETIC FIELD
• MUTUAL INDUCTANCE
• SELF-INDUCTANCE
• ENERGY STORED IN MAGNETIC FIELD
• ALTERNATING CURRENT AND AC-CIRCUITS
• ALTERNATING CURRENT
• LR-CIRCUITS.....................................................................................................................................

Electrostatics

Electric charge and electric field

There are two types of electric charge – positive and negative. The interaction between charges of charges particles of bodies depend on what is the type of charge that each charge/particle/body have. The interaction occurs according to the following rule Like charges repel, unlike charges attract. Each type of charge is assigned a sign, that is one of them is positive (has + sign), another one is negative (has – sign). That is, if two charges are of the same sign , they repel, fi they are of the opposite signs , they attract. The sign has to be treated algebraically during calculations, that is, negative charge has to have negative value etc. During studies of charges it was detected, that no charge can disappear or appear from nowhere. This resulted in formulation of the law of conservation of charge. The net amount of electric charge produced in any process is zero. or No net electric charge can be created or destroyed.

Electric field

Electric field is defined to easily describe interaction of several charges. Since we know that charges can repel and attract, we want to know the force exerted on some charge q 1 placed near some another charge Q. Even more, we want to know the force exerted by Q on another charge q 2. That means, it would be very convenient to define electric field as something that does not depend on charges that we are placing near Q. Thus, to define electric field we use something called a test charge. A test charge is such a small charge placed near charge Q that is does not significantly affect the charge Q itself. That means, a test charge must be so small, it exerts neglectable force on Q. Based on that definition, we define electric field as follows: electric field is a force exerted on a tiny positive test charge placed at a given point, divided by the magnitude of that test charge. 𝐸^1 ⃗^ =

• ⃗ / [definition of electric field] (1.4) More precisely, electric field is defined as limit of Eq. 1.4 when q approaches zero. That means, q is so small, it has no influence of charges that create the field. By doing that we achieve to define the effect (force) of the charges that produce electric field without any link to the charges placed into that field. Imagine a positive charge Q that produces electric field and several tiny (test) charges placed near Q. Test charges are positive, so they repel and Coulomb’s force points away from Q. From definition of electric field and Eq. 1.1, we can write electric field for each of those charges A, B and C. 𝐸^1 ⃗^ =
• ⃗ /

(/!/"" /

! ""^ [for point charge] (1.5) Writing Eq. 1.5 in terms of Î 0 , 𝐸^1 ⃗^ =

$ pÎ# ! ""^ [field for point charge in terms of Î 0 ] (1.6) We can obtain the force exerted on a test charge by electric field rearranging Eq. 1.4. 𝐹⃗ = 𝑞𝐸^1 ⃗^ (1.7) That equation is valid if q is small enough not to move charges that produce the field.

Field lines

It is useful to visualize the electric field even when there is no test charges present. We could use arrows for that, however, since electric field expands in all directions, that would become confusing due to the number of arrows. Thus, we use field lines instead. Field lines have the following properties

1. Electric field lines indicate direction if the electric field. The field points in the direction tangent to the field line at any point.
2. The lines are drawn so that the magnitude of the electric field is proportional to the number of lines crossing unit area perpendicular to the lines. (The closer together the lines, the stronger the field)
3. Electric field lines start on positive charges are end on negative charges; the number of starting and ending lines is proportional to the magnitude of the charge. Electric field lines start at positive charge and end on negative charge (property 3) Since the area of a sphere 2 is larger than area of sphere 1, the magnitude of electric field at distance to sphere 1 is larger than that at distance to sphere 2 (property 2) The electric field (and thus the force exerted on an electric charge placed in that field) points in the direction tangent to the field line (property 1). Note that the number of electric field lines that starts/ends on an electric charge is proportional to the magnitude of that charge. That means, if n lines leave particle of charge q , then 2n lines will leave particle of charge 2q.

We write torque with minus sign since its direction (rotation) is opposite to direction of increasing q.

𝑊 = ∫ t 𝑑 q

q"

q! =^ −𝑝𝐸^ ∫^ 𝑠𝑖𝑛^ q^ 𝑑^ q

q"

q! =^ 𝑝𝐸(cos^ q#^ −^ cos^ q&)^ (1.12)

Positive work done by electric field on dipole decreases its potential energy. If we choose U = 0 when 𝑝⃗ || 𝐸^1 ⃗^ (that is q 1 = 90o) and then setting q 2 = q, we get^1

𝑈 = −𝑝⃗ × 11 𝐸^1 ⃗^ [uniform electric field] (1.13)

(^1) Recall DU = - W, that is change in potential energy is equal to the negative work done.

Gauss’ law

Coulomb’s law is convenient when we have one or several charges. however, when we have many of them, determining electric field around them becomes complicated, because electric field will be equal to the vector sum of all the individual fields produced by each charge. Instead we use the Gauss’ law and the concept of flux that helps to deal with any amount of charge as long as it is enclosed into some area.

Electric flix

Electric field passing through a given area is called an electric flux. This is also referred as number of electric field lines passing through a given area. For a uniform electric field passing through an area A (as shown), electric flux is defined as

F 1 = 𝐸𝐴𝑐𝑜𝑠 q (2.1)

Where q is the angle between the electric field direction and the line drawn perpendicular to the area. For area A represented as a vector pointing perpendicular to the surface and having magnitude A, electric flux can be written in a vector form.

F 1 = 𝐸^1 ⃗ × 𝐴⃗ (2.2)

As stated, electric flux refers to the number of electric field lines passing through the area A.

F 1 μ 𝑁 (2.3)

Where N refers to the number of field lines passing through area A. When the electric field is not uniform or/and the surface is not flat, we divide the surface into infinitesimal area that can be considered flat and electric field though which can be considered uniform and integrate over that surface.

F 1 = ∫ 𝐸^1 ⃗^ 𝑑𝐴⃗ [electric flux in general form] (2.4)

Area A

Electric potential

Electric potential energy and potential difference

To define an electric potential energy, we need a conservative force^1. We are lucky (or not), because Coulomb’s force, that is, force of attraction/repulsion of charges is conservative. Now, recall that potential energy is equal to the negative of the work done.

D𝑈 = −𝑊 (3.1)

Electric potential energy If we place a point charge in a uniform electric field, for example between two long charged plates, and release that charge, it will move in a direction that results in reduction of potential energy. The work done by electric field to move point charge from positive plate to negative plate is 𝑊 = 𝐹𝑑 = 𝑞𝐸𝑑 (3.2) Since change in potential energy is negative of work done (Eq. 1.20), the change in potential energy in a uniform electric field will be

D𝑈 = −𝑞𝐸𝑑 (3.3)

(^1) The force is called conservative if work done by that force does not depend on the path chosen, but only depend on initial and final positions (reminds a bit of state functions in thermodynamics). The most common example of a conservative force is gravitational force: the work done by falling object does not depend on it’s path, but only on height.

Electric potential Again, as with electric field, it is very convenient to determine electric potential of any charge just knowing its position relative to some other charge. That is, we want (again) to get rid of charge that influences potential energy change to determine potential energy of any charge at a given point knowing only one values (should have reminded you of electric field) Thus, we define electric potential as electric potential energy difference per unit charge. 𝑉 2 = (^3) * /

However, only the potential difference , is useful (and possible to measure), that is we also define the potential difference.

D𝑉 =

(^3) +) (^3) * /

(^5) +* /

With that definition we can determine potential energy of every charge knowing it’s position relative to the charge producing electric field and knowing the electric potential in that point.

D𝑈 = 𝑞 D𝑉 (3.6)

Since Eq. 1.23 units of electric potential are volt , 1V = 1 J/C. The difference in potentials is usually referred as voltage.

Electric potential due to several charges Since electric field obeys superposition principle, when we have several charges, we can just sum up their contribution to the potential in a given point. Since assumption of V = 0 at r = ¥ is very convenient, we that with that and write for n point charges 𝑉 = ∑^879 # 𝑉 7 =

$ pÎ#

!, ",* 8 79 # (3.12) Where ria is a distance from the ith^ charge Qi to the point a. In case charge distribution is continuous, we can write 𝑉 =

$ pÎ#

:/ " [continuous charge distribution] (3.13)

Electric dipole potential

The electric potential at an arbitrary point P due to a dipole is the sum of the potentials due to each of the two charges (again V= 0 at r = ¥). For an arbitrary point P, a distance r from charge +Q and distance r+Dr from charge - Q of the dipole, the sum of electric potentials is 𝑉 =

$ pÎ# ! "

$ pÎ# ()!) ("= D") Or 𝑉 = ! $ pÎ# D" "("= D") [V = 0 at r = ¥] (3.14) As we move away from the dipole, the distance l between two charges becomes neglectable in comparison with distance r (r >> l). From picture we can see that

D𝑟 » 𝑙 cos q (3.15)

And since r >> Dr = l cosq, we can neglect Dr in denominator and obtain 𝑉 =

$ pÎ# > ?@A q ""^ [recall Eq. 1.8, for r >> l , dipole] (3.16) Sometimes (in chemistry) dipoles are used to describe molecules and unit of Coulomb-meter (C*m) is too large, so Debye is used instead.

1 𝑑𝑒𝑏𝑦𝑒 = 3. 33 ∗ 10 )B'^ 𝐶 × 𝑚 [Debye unit] (3.17)