Topic 2: Charges on conductors and electrostatic conditions. Electric field at the surfaces of a conductor; charging by induction môn Vật lý đại cương 2 | Học viện Công Nghệ Bưu Chính Viễn Thông

Part I: 20 topics
Topic 1: Electric flux and Gauss’s law. Applications of Gauss’s law on
planar, spherical and cylindrical distribution. 1. Definition: a. Electric flux:
Electric flux is a measure of the “flow” of electric field through a surface. It is equal to the
Φ𝐸= ∫ 𝐸 cos 𝜑 𝑑𝐴 = ∮ 𝐸󰇍 𝑑𝐴
product of an area element and the perpendicular component of E, integrated over a surface. b. Gauss's law: 𝑑𝐴 =𝑄𝑒𝑛𝑐𝑙
Gauss's law state that the total electric flux through a closed surface, which can be
written as the surface integral of the component of E normal to the surface, equals a constant
Φ𝐸= ∫ 𝐸 cos 𝜑 𝑑𝐴 = ∮ 𝐸󰇍 𝜖0
times the total charge Qencl enclosed by the surface.
2. Applications of Gauss’s law
If we know the charge distribution, and if it has enough symmetry to let us evaluate the integral
in Gauss’s law, we can find the field. 𝐸 = 𝜎
a. Planar: infinite sheet of charge with uniform charge per unit area 𝜎
If we know the field, we can use Gauss’s law to find the charge distribution, such as charges on conducting surfaces. 2𝜀0
b. Spherical: Solid insulating sphere with radius R, charge Q distributed uniformly 𝐸 = 𝑄 throughout volume: 4𝜋𝜀0𝑟2
Outside sphere, r>R: E=Q/4pi e0 r square 𝐸 = 𝑄𝑟 4𝜋𝜀0𝑅3 Inside sphere, r
Chagre q on the surface of conducting sphere with radius R:
outside sphere, r>R: E=q/4pi e0 r square 𝐸 = 𝑞4𝜋𝜀0𝑟2
inside sphere, rc. Cylindrical: infinite conducting cylinder with radius R, charge per unit length lamda:
Outside cylinder, r>R: 𝐸 = 𝜆2𝜋𝜀0
Inside cylinder, rTopic 2: Charges on conductors and electrostatic conditions. Electric
field at the surfaces of a conductor; charging by induction.
1. Charges on conductor:
A conductor allows free charges to move about within it. The electrical force around a
conductor will cause free charges to move around inside the conductor until static equilibrium
is reach. Any excess charge will reside entirely on the surface, not in the interior of the material.
2. Electrostatic conditions:
The charges have no net motion. Giving a solid conductor with charge q then the charge q
resides entirely on the surface of the conductor. The situation is electrostatic, so E=0 within the conductor.
3. Electric field at the surfaces of a conductor:
There is a direct relationship between the E field at a point just outside any conductor and the
surface charge density o at that point.
E normal A=oA/e0 and e normal= o/e0
4. Charging bay induction:
Induction charging is a method used to charge an object without actually touching the object to any other charged object.
Topic 3: Electric potential and electric potential difference. Calculating
the electric potential. Electric potential energy, equipotential surfaces. 1. Definition: a. Electric potential
Electric potential at any point in an electric field is potential energy U per unit charge associated
with a test charge q0 at that point, denoted by V: 𝑉 = 𝑈𝑞0 or U=𝑞0V
b. Electric potential difference
Electric potential difference between 2 points a and b, also called the potential of a with
respect to b, equals the work done by the electric force when a unit charge move from a to b. It
is given by the line integral of E: Va-Vb=integral E dl
𝑉𝑎− 𝑉𝑏= ∫ 𝐸 󰇍 𝑑𝑙
2. Calculating the electric potential: a. Due to point charge: 𝑉 = 𝑞 4𝜋𝜀0𝑟
b. Due to a collection of point charges 𝑉 = 14𝜋𝜀0∑𝑞𝑖 𝑖 𝑟𝑖 c. Due to charge distribution
𝑉 = 14𝜋𝜀0∫𝑑𝑞𝑟
3. Electric potential energy:
The electric potential energy is the work W done by the electric force on a charged particle moving in an electric field. U=qq0/4pi e0 r 𝑈 = 𝑞𝑞0 4𝜋𝜀0𝑟
4. Equipotential surfaces:
An equipotential surface is a surface on which the potential has the same value at every
point. At a point where a field line crosses an equipotential surface, the two are
perpendicular. When all charges are at rest, the surface of a conductor is always an
equipotential surface and all points in the interior of a conductor are at the same potential.
When a cavity within a conductor contains no charge, the entire cavity is an equipotential
region and there is no surface charge anywhere on the surface of the cavity.
Topic 4: Relations between electric field vector and electric potential, potential gradien
If the potential is known as a function V
of the coordinates x, y, and z, the components of
electric field 𝐸󰇍 at any point are given by partial derivatives of V.
𝐸𝑥= − 𝜕𝑉𝜕𝑥 𝐸𝑦= − 𝜕𝑉𝜕𝑦 𝐸𝑧= − 𝜕𝑉𝜕𝑧 𝐸󰇍 = −(𝑖 𝜕𝑉 𝜕𝑥 + 𝑗 𝜕𝑉 𝜕𝑦 + 𝑧 𝜕𝑉 𝜕𝑧)
𝐸󰇍 is the negative of the gradient of V 𝐸󰇍 = −∇󰇍 𝑉
Topic 5: Conductors, Capacitors, Capacitance of parallel-plate and
cylindrical capacitors; Energy Storage in Capacitors and Electric-Field Energy.
1. Conductor: materials that permit the easy movement of charge through them
2. Capacitor: A capacitor is any pair of conductors separated by an insulating material (or a vacuum) 3. Capacitance:
a. Of parallel-plate capacitor: 𝐶 = 𝑄𝑉𝑎𝑏 =𝜖𝑜 𝐴𝑑
b. Of cylindrical capacitor: 𝐶 = 𝑄𝑉𝑎𝑏 =𝜆𝐿 𝜆
2𝜋𝜖0ln𝑟𝑏𝑟𝑎 =2𝜋𝜖0𝐿 ln(𝑟𝑏 𝑟𝑎)
4. Enenrgy storage in capacitors and electric field energy: 𝑈 = 𝑄2 2𝐶 =12𝐶𝑉2=12𝑄𝑉
Topic 6: Magnetic field, magnetic field lines, magnetic flux. Motion of
charged particles in a magnetic field. 1. Magnetic field:
A moving charge or a current creates a magnetic field in the surounding space.
2. Magnetic field lines:
We can represent any magnetic field by magnetic field lines. We draw the lines so that
the line through any point is tangent to the magnetic field vector 𝐵󰇍 at that point. Also,
because the direction of 𝐵󰇍 at each point is unique, field lines never intersect. 3. Magnetic flux:
Magnetic flux is a measure of the “flow” of magnetic field through a surface. It is equal
to the product of an area element and the perpendicular component of B, integrated over a surface.
𝑑Φ𝐵= 𝐵𝑐𝑜𝑠𝜃𝑑𝐴 = 𝐵󰇍 ∙ 𝑑𝐴
Φ𝐵= ∫𝐵󰇍 ∙ 𝑑𝐴
4. Motion of charged particles in a magnetic field.
Motion of a charged particle under the action of a magnetic field alone is always motion with constant speed. 𝑅 = 𝑚𝑣 |𝑞|𝐵
Topic 7: Magnetic forces of currents, concept of magnetic field.
Magnetic field vector; magnetic field of a current element; the law of
Biot and Savart, magnetic field of a moving charge.
1. Magnetic force of currents:
A straight segment of a conductor carrying current in a uniform magnetic field
𝐵󰇍 experiences a force 𝐹󰇍 that is perpendicular to both 𝐵󰇍 and the vector 𝑙 which points in
the direction of the current and has magnitude equal to the length of the segment.
𝐹 =𝐼𝑑𝑙 × 𝐵 󰇍
2. Concept of magnetic field:
A magnetic field is a vector field which is created by a moving charge or a current exerts
a force on any other moving charge or current that is presented in the field.
3. Magnetic field vector and magnetic field of a current element:
- Magnetic field of a current element: The total magnetic field caused by several moving
charges is the vector sum of all fields caused by individual charges.
-At each point, the magnetic field vector is always tangential to the magnetic field line crossing that point.
4. The law of Biot and Savart:
The law of Biot and Savart gives the magnetic field 𝑑𝐵󰇍 created by an element of a
conductor 𝑑𝑙 carrying current I. The field d𝐵󰇍 is perpendicular to both 𝑑𝑙 and 𝑟 the unit
vector from the element to the field point. The 𝐵
󰇍 field created by a finite current-
carrying conductor is the integral of over the length of the conductor.
𝑑𝐵󰇍 =𝜇04𝜋 𝐼𝑑𝑙 × 𝑟 𝑟2
5. Magnetic field of a moving charge: 𝐵󰇍 =𝜇04𝜋𝑞𝑣 × 𝑟 𝑟2
Topic 8: Ampere’s law on the line integral of magnetic field vector
around a closed path. Applications of Ampere’s law in calculating the
magnetic field of currents.
1. Ampere’s law:
Ampere’s law states that the line integral of 𝐵
󰇍 around any closed path equals times the
net current through the area enclosed by the path. The positive sense of current is
determined by a right-hand rule.
∮ 𝐵󰇍 ∙ 𝑑𝑙 = 𝜇0𝐼𝑒𝑛𝑐𝑙 2. Application: Current Point in Magnetic-Field Magnitude distribution magnetic field Long, straight Distance r from 𝐵 = 𝜇0𝐼 conductor conductor 2𝑟𝜋 Circular loop of On axis of loop 𝐵 = 𝜇0𝐼𝑎2 radius a 2(𝑥2+ 𝑎2)3/2 At center of loop 𝐵 = 𝜇0𝐼 (for N loops, 2𝜋𝑎 multiply these expressions by N) Long cylindrical Inside conductor, 𝐵 = 𝜇0𝐼𝑟 conductor of radius r2𝜋𝑅2 R Outside conductor, 𝐵 = 𝜇0𝐼 r>R 2𝑟𝜋 Long, closely wound Inside solenoid, 𝐵 = 𝜇0𝑛𝐼 solenoid with n near center turns per unit Outside solenoid 𝐵 ≈ 0 length, near its midpoint Tightly wound Within the space 𝐵 = 𝜇0𝑁𝐼 toroidal solenoid enclosed by the 2𝑟𝜋 with N turns windings, distance r from symmetry axis Outside the space 𝐵 ≈ 0 enclosed by the windings
Topic 9: Induction Experiments, Faraday’s Law, Lenz’s Law
1. Induction Experiments: When we move the magnet either toward or away from the coil,
the meter shows current in the circuit, but only while the magnet is moving. If we keep
the magnet stationary and move the coil, we again detect a current during the motion.
2. Faraday’s law:
Faraday’s law states that the induced emf in a closed loop equals the negative of the
time rate of change of magnetic flux through the loop. This relationship is valid whether
the flux change is caused by a changing magnetic field, motion of the loop, or both. 𝜀 = −𝑑Φ𝐵 𝑑𝑡
3. Lenz’s law:
Lenz’s law states that an induced current or emf always tends to oppose or cancel out the change that caused it.
Topic 10: Motional Electromotive Force; Induced Electric Fields; Eddy Currents
1. Motional electromotive force: a. The rod:
The moving rod, which slides along a stationary U- shaped conductor, has become a
source of electromotive force, charge move from lower to higher potential. 𝜀 = 𝑣𝐵𝑙
b. The closed conducting loop:
For an element 𝑑𝑙 of the conductor, the contribution 𝑑𝜀 to the emf is the magnitude
dl multiplied by the component of 𝑣 × 𝐵󰇍 (the magnetic force per unit charge)
parallel to 𝑑𝑙 ; that is, 𝑑𝜀 = 𝑣 × 𝐵 󰇍 ∙ 𝑑𝑙
For any closed conducting loop, the total emf is:
𝜀 = ∮ 𝑣 × 𝐵󰇍 ∙ 𝑑𝑙
2. Induced electric field: there is an induced electric field in the conductor caused by the changing magnetic flux
∮ 𝐸󰇍 ∙ 𝑑𝑙 = − 𝑑𝐵 𝑑𝑡
3. Eddy current: Many pieces of electrical equipment contain masses of metal moving in magnetic
fields or located in changing magnetic fields. In this situations like these we can have induced
currents that circulate throghout the volume of a material, we cal these eddy currents.
Topic 11: Mutual Inductance, Self-Inductance and Inductors. Energy
stored in a inductor, magnetic field energy, magnetic energy density 1. Mutual inductance:
If the circuits are coils of wire with 𝑁1 and 𝑁2 turns, the mutual inductance M can be
expressed in terms of the average flux Φ𝐵2 through each turn of coil 2 caused by the
current 𝑖1 in coil 1, or in terms of the average flux Φ𝐵1through each turn of coil 1 caused
by the current 𝑖2 in coil2. 𝜀2 = −𝑀 𝑑𝑖1
𝑑𝑡 𝜀1 = −𝑀 𝑑𝑖2 𝑑𝑡 𝑀 = 𝑁2Φ 𝑖1𝐵2 =𝑁1Φ𝐵1𝑖2
2. Self-inductance: A changing current I in any circuit causes a self-induced ε. The
inductance (or self-inductance) L depends on the geometry of the circuit and the
material surrounding it. The inductance of a coil of N turns is related to the average flux
Φ𝐵 through each turn caused by the current I in the coil. 𝜀 = −𝐿 𝑑𝑖 𝑑𝑡 𝐿 = 𝑁Φ𝐵 𝑖 3. Inductor:
It is a circuit device, usually including a coil of wire, intended to have a substantial inductance.
4. Magnetic-field energy:
An inductor with inductance L carrying current I have energy U associated with the inductor’s magnetic field 𝑈 = 12𝐿𝑖2
5. The magnetic energy density u (energy per unit volume) is proportional to the square of the magnetic field magnitude 𝑢 = 𝐵2
2𝜇0(𝑖𝑛 𝑣𝑎𝑐𝑢𝑢𝑚) 𝑢 = 𝐵2
2𝜇 (𝑖𝑛 𝑎 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑤𝑖ℎ 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝜇)
Topic 12: R-L-C circuit, damped harmonic motion 1. R-L-C circuit:
It is a circuit that contains inductance, resistance, and capacitance?????
2. Damped harmonic motion:
A circuit that contains inductor, resistor, and capacitor and undergoes damped
oscillation for sufficiently small resistance. The frequency w’ of damped oscillations
depends on the value of L, R, and C. As R increases, the damping increases; if R is greater
than a certain value, the behavior becomes overdamped and no longer oscillates. 𝜔′=√1𝐿𝐶 −𝑅2 4𝐿2
Topic 13: The system of Maxwell’s equations, the concept of electromagnetic field
1. The system of Maxwell’s equations: a. Gauss’s law:
∮ 𝐸󰇍 ∙ 𝑑𝐴 =𝑄𝑒𝑛𝑐𝑙∈0
b. Gauss’s law for magnetism: ∮ 𝐵󰇍 ∙ 𝑑𝐴 = 0 c. Ampere’s law: ∮ 𝐵
󰇍 ∙ 𝑑𝑙 = 𝜇0(𝑖𝐶+∈0dΦ𝐸𝑑𝑡 )𝑒𝑛𝑐𝑙 d. Faraday’s law:
∮ 𝐸󰇍 ∙ 𝑑𝑙 = − 𝑑Φ𝐸 𝑑𝑡
2. The concept of eletromagneticfield:
Electromagnetic field, a property of space caused by the motion of an electric charge. A
stationary charge will produce only an electric field in the surrounding space. If the
charge is moving, a magnetic field is also produced. An electric field can be produced
also by a changing magnetic field.
Topic 14: Derivation of the wave equation Electromagnetic Wave,
sinusoidal Plane Electromagnetic Waves, polarization and the Speed of Light
1. Derivation of the wave equation: 3. Speed of light:
Topic 15: Kinetic-molecular model of an ideal gas. Average
translational kinetic energy of a gas molecule
1. Kinetic-molecular model of an ideal gas:
a. A container with volume V contains a very large number N of identical molecules, each with mass m.
b. The molecules behave as point particles that are small compared to the size ò the
container and to the average distance between molecules.
c. The molecules are in constant motion. Each molecule collides occasionally with a wall
of the container. These collisions are perfectly elastic.
d. The container walls are rigid and infinitely massive and do not move.
2. Average translational kinetic energy of a gas molecule
The average translational kinetic energy per molecule depends only on temperature, not on the
pressure, volume, or kind of molecule 12𝑚𝑣 2=3 𝑎𝑣 2𝑘𝑇
Topic 16: Distribution function of molecular speeds, the Maxwell-
Boltzmann distribution. Average speed, root-main square speed and the
most probable speed of a gas molecule
1. Distribution function of molecular speeds:
If we observe a total of N molecules, the number dN having speeds in the range
between v and v+dv is given by: dN = Nf(v)dv
We can also say that the probability that a randomly chosen molecule will have a speed
in the interval v +dv is f(v)dv. Hence f(v) is the probability per unit speed interval.
2. The Maxwell-Boltzmann distribution: 𝑓(𝑣)= 4𝜋( 𝑚
2𝜋𝑘𝑇)3/2𝑣2𝑒−𝑚𝑣2/2𝑘𝑇 3. Average speed:
𝑣𝑎𝑣 =√8𝑘𝐵𝑇 𝜋𝑚 = √8𝑅𝑇 𝜋𝑀
4. Root- mean- square speed 𝑣
2=√3𝑘𝑇𝑚=√3𝑅𝑇 𝑟𝑚𝑠 =√𝑣𝑎𝑣 𝑀
5. Most probable speed: 𝑣𝑚𝑝 =√2𝑘𝐵𝑇 𝑚=√2𝑅𝑇𝑀
Topic 17: The first law of thermodynamics and applying for kinds of
thermodynamic processes: adiabatic, isochoric, isobaric and isothermal processes:
1. The first law of thermodynamics:
States that when heat Q is added to a system does work W, the internal energy U
changes by an amount equal to Q – W. This law can also be expressed for an infinitesimal process. ∆𝑈 = 𝑄 − 𝑊
𝑑𝑈 =𝑑𝑄 −𝑑𝑊
2. Applying for kinds of thermodynamic processes:
a. Adiabatic: is defined as one with no heat transfer into or out the system; Q=0
𝑈2− 𝑈1= ∆𝑈 = −𝑊
b. Isochoric: is a constant-volume process. When the volume of a thermodynamic
system is constant, it does no work on it surroundings. Then W = 0 and
𝑈2− 𝑈1= ∆𝑈 = 𝑄
c. Isobaric process: is a constant-pressure process. In general, none of the three
quantities ∆𝑈, 𝑄 𝑎𝑛𝑑 𝑊 is zero in an isobaric process, but calculating W is easy nonetheless. 𝑊 = 𝑝(𝑉2− 𝑉1)
d. Isothermal process: is a constant-temperature process. For a process to be
isothermal, any heat flow into or out the system must occur slowly enough that
thermal equilibrium is maintained. In general, none of the quantities ∆𝑈, 𝑄 𝑎𝑛𝑑 𝑊
is zero in an isothermal process.
Topic 18: Directions of thermodynamic processes, reversible and
irreversible processes. Heat engines, thermal efficiency of an engine the
Carnot cycle and quantitative expression of the second law of thermodynamic:
1. Directions of thermodynamic processes:
Thermodynamic processes that occur in nature are all irreversible processes. These are
processes that proceed spontaneously in one direction but not the other. The flow of
heat from a hot body to is irreversible, as is the free expansion of a gas.
2. Reversible processes: are equilibrium processes, with the system always in
thermodynamic equilibrium. Of course, if a system were truly in thermodynamic
equilibrium, no change of state would take place.
3. Irreversible processes: are heat flow with finite temperature difference, free expansion
of a gas, and conversion of work to heat; no small change in conditions could make any
of them go the other way. They are also all nonequilibrium processes, in that the system
is not in thermodynamic equilibrium at any point until the end of the process.
4. Thermal efficiency of an engine is the quotient 𝑒 = 𝑊
𝑄𝐻= 1 + 𝑄𝐶𝑄𝐻= 1 − |𝑄𝐶 𝑄𝐻|
W: the net work done by the working substance.
𝑄𝐻 : the amount of heat energy by the hot reservoir (>0)
𝑄𝐶: the heat rejected in the exhaust (<0)
5. The Carnot cycle is the hypothetical, idealized heat engine that has the maximum
possible efficiency consistent with the second law. It consists of the following steps:
+ The gas expands isothermally at temperature 𝑇𝐻 absorbing heat 𝑄𝐻
+ It expands adiabatically until its temperature drops to 𝑇𝐶
+ It is compressed isothermally at 𝑇𝐶 , rejecting heat |𝑄𝐶|
+ It is compressed adiabatically back to its initial state at temperature 𝑇𝐻
6. Quantitative expression of the second law thermodynamics:
It is impossible for any system to undergo a process in which it absorbs heat from a
reservoir at a single temperature and converts the heat completely into mechanical
work, with the system ending in the same state in which it began.
Topic 19: Entropy change in reversible and irreversible processes, the
meaning of the second law of thermodynamics: 2
1. Entropy changes in reversible processes: ∆𝑆 = ∫𝑑𝑄 1 𝑇
When a system proceeds from an initial state with entropy S1 to a final state with entropy S,
the change in the entropy ∆𝑆 = 𝑆2− 𝑆1 defined by the above equation does not depend on the
path leading from state 1 to state2.
2. Entropy changes in irreversible processes: all irreversible processes involve an increase in
entropy. The entropy of an isolated system can change, but it can never decrease.
The second law limits the availability of energy and the ways in which it can be used and converted.
Topic 20: Differences between ideal and real gases, Van der Waals
equation of state, critical point:
1. Differences between ideal and real gas:
Difference between Ideal gas and Real gas IDEAL GAS REAL GAS No definite volume Definite volume
Elastic col ision of particles
Non-elastic col isions between particles
No intermolecular attraction forces
Intermolecular attraction force
Does not real y exists in the environment and is a
It real y exists in the environment hypothetical gas High pressure
The pressure is less when compared to Ideal gas Independent Interacts with others Obeys PV = nRT
Obeys p + ((n2 a )/V2)(V – n b ) = nRT
2. Van der Waals equation of state: (𝑝 + 𝑎𝑛2𝑉2)(𝑉 − 𝑛𝑏)= 𝑛𝑅𝑇 3. Critical point:
The endpoint at the top of the vaporization curve, at which distinction between liquid and vapor disappears. Part II: Final topics 2013
Topic 1: Gauss's law for the case of electric field and magnetic field: 1. For electric field:
Gauss's law state that the total electric flux through a closed surface, which can be written as
the surface integral of the component of E normal to the surface, equals a constant time the
total charge Qencl enclosed by the surface.
Φ𝐸= ∫ 𝐸 cos 𝜑𝑑𝐴 = ∮ 𝐸 󰇍 𝑑𝐴 =𝑄𝑒𝑛𝑐𝑙𝜖0 2. For magnetic field:
The total magnetic flux through a closed surface is always zero. ∮ 𝐵󰇍 ∙ 𝑑𝐴 = 0
Topic 2: The relationship between electric field and potential, prove that
vector E normal to the equipotential lines:
1. The relationship between electric field and potential:
If the potential V is known as a function of the coordinates x, y, and z, the components of
electric field 𝐸󰇍 at any point are given by partial derivatives of V.
𝐸𝑥= − 𝜕𝑉𝜕𝑥 𝐸𝑦= − 𝜕𝑉𝜕𝑦 𝐸𝑧= − 𝜕𝑉𝜕𝑧 𝐸󰇍 = −(𝑖 𝜕𝑉 𝜕𝑥 + 𝑗 𝜕𝑉 𝜕𝑦 + 𝑧 𝜕𝑉 𝜕𝑧)
2. 𝐸󰇍 is the negative of the gradient of V 𝐸󰇍 = −∇󰇍 𝑉
Topic 3: The energy and energy density of electric field:
1. The energy of electric field:
A capacitor with capacitance C carrying voltage V has energy U associated with the capacitor’s electric field 𝑈 = 𝑄2 2𝐶 =12𝐶𝑉2=12𝑄𝑉
2. The energy density of electric field:
It is the energy per unit volume in the space between the plates of a parallel-plate
capacitor with plate area and separation d
𝑢 = 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = 12𝐶𝑉2 𝐴𝑑 =1 2𝜖0𝐸2
Topic 4: Ampere’s law and applications:
1. Ampere’s law:
Ampere’s law states that the line integral of 𝐵
󰇍 around any closed path equals times the net
current through the area enclosed by the path. The positive sense of current is determined by a right-hand rule.
∮ 𝐵󰇍 ∙ 𝑑𝑙 = 𝜇0𝐼𝑒𝑛𝑐𝑙 2. Application: Current Point in Magnetic-Field Magnitude distribution magnetic field Long, straight Distance r from 𝐵 = 𝜇0𝐼 conductor conductor 2𝑟𝜋 Circular loop of On axis of loop 𝐵 = 𝜇0𝐼𝑎2 radius a 2(𝑥2+ 𝑎2)3/2 At center of loop 𝐵 = 𝜇0𝐼 (for N loops, 2𝜋𝑎 multiply these expressions by N) Inside conductor, 𝐵 = 𝜇0𝐼𝑟 r2𝜋𝑅2 Long cylindrical Outside conductor, 𝐵 = 𝜇0𝐼 conductor of radius r>R 2𝑟𝜋 R Long, closely wound Inside solenoid, 𝐵 = 𝜇0𝑛𝐼 solenoid with n near center turns per unit Outside solenoid 𝐵 ≈ 0 length, near its midpoint Tightly wound Within the space 𝐵 = 𝜇0𝑁𝐼 toroidal solenoid enclosed by the 2𝑟𝜋 with N turns windings, distance r from symmetry axis Outside the space 𝐵 ≈ 0 enclosed by the windings
Topic 5: Faraday’s law and self inductance:
1. Faraday’s law:
Faraday’s law states that the induced emf in a closed loop equals the negative of the time rate
of change of magnetic flux through the loop. This relationship is valid whether the flux change is
caused by a changing magnetic field, motion of the loop, or both. 𝜀 = − 𝑑Φ𝐵 𝑑𝑡 2. Self inductance:
A changing current I in any circuit causes a self-induced ε. The inductance (or self-inductance) L
depends on the geometry of the circuit and the material surrounding it. The inductance of a coil
of N turns is related to the average flux Φ𝐵 through each turn caused by the current I in the coil. 𝜀 = −𝐿 𝑑𝑖 𝑑𝑡 𝐿 = 𝑁Φ𝐵 𝑖
Topic 6: Energy and energy density of magnetic field
1. Magnetic-field energy:
An inductor with inductance L carrying current I has energy U associated with the inductor’s magnetic field 𝑈 = 12𝐿𝑖2
2. The magnetic energy density u (energy per unit volume) is proportional to the square of the magnetic field magnitude 𝑢 = 𝐵2
2𝜇0(𝑖𝑛 𝑣𝑎𝑐𝑢𝑢𝑚) 𝑢 = 𝐵2
2𝜇 (𝑖𝑛 𝑎 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑤𝑖ℎ 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝜇)
Topic 7: Set up the equation for a damping electromagnetic oscillation in a LCR circuit:???
A circuit that contains inductance, resistance, and capacitance and undergoes damped oscillation for
sufficiently small resistance. The frequency w’ of damped oscillations depends on the value of L, R, and
C. As R increases, the damping increases; if R is greater than a certain value, the behavior becomes
overdamped and no longer oscillates. 𝜔′=√1𝐿𝐶 −𝑅2 4𝐿2
Topic 8: From Maxwell’s statement (based on Pharaday’s and
Amprere’laws) set up the equations for electromagnetic wave: ????
Topic 9: Using the principle of equipartition of energy derive the
expression of internal energy for monatomic, diatomic and polyatomic ideal gases:???
Topic 10: Express the statement of 1st law of thermodynamics and drive
the expressions of 4 basic processes of ideal gas:
1. The first law of thermodynamics:
States that when heat Q is added to a system does work W, the internal energy U changes by an
amount equal to Q – W. This law can also be expressed for an infinitesimal process. ∆𝑈 = 𝑄 − 𝑊
𝑑𝑈 =𝑑𝑄 −𝑑𝑊
2. Applying for kinds of thermodynamic processes:
a. Adiabatic: is defined as one with no heat transfer into or out the system; Q=0
𝑈2− 𝑈1= ∆𝑈 = −𝑊