Physics comprehensive

It is assumed that the original particle decays while at rest. Conservation of momentum requires that m and the massless particle have oppositely directed, equal magnitude (p) momenta in the final state. Energy conservation requires that: .

The three basic passive elements are the capacitor, resistor and inductor. Each is characterized by its voltage rule. V_C =( ¹/_C) Q ; ­ V_R = I R ; V_L = L ^dI/_dt (Note ¹/_C in the V_C relation.)

Series Combos: R_seires= R₁ + R₂ L_series= L₁ + L₂

Parallel Combos: C_parallel = C₁ + C₂

Study the location C relative to those of R and L in the voltage relations. Note the inverse location

of C in the voltage relations and hence in the rules for series and parallel combinations.

The voltage across a capacitor is a continuous function of time. Why?

The current through an inductor is a continuous function of time. Why?

Kirchhoff’s voltage rule: The sum of the voltages (potential changes) around a closed path is zero.

Kirchhoff’s current rule: The algebraic sum of the currents into a node is zero.

Elements are in series if the wiring of the circuit requires that the current through the elements is the same. There can be no branch points between them.

The voltage across a series combination is the sum of the voltages across the individual elements.

Elements are in parallel if the wiring of the circuit requires that the voltage across the elements is the same. One lead from each goes to a common point and the other two leads from the elements join at another common point (connection path to join of high conductivity).

The current through a parallel combination is the sum of the currents through the individual elements.

SERIES PARALLEL

http://en.wikipedia.org/wiki/File:Parallel_circuit.svg

Elements are in parallel if the wiring of the circuit requires that the voltage across the elements is the same. One lead from each goes to a common point and the other two leads from the elements join at another common point (connection path to join of high conductivity).

Ex. 8) A capacitor of capacitance 125 microfarads is initially charged such that the voltage across the plates is found to be 100 volts. If the capacitor is the connected in series to (with) a pure inductor of inductance 0.2 Henry, what is the maximum value of the current that is observed in the inductor? (A) 1.0 A (B) 2.5 A (C) 4.5 A (D) 5.0 A (E) 10.0 A

Prepare a sketch: L C 100 V 125 F 0.2 H

When the switch is closed, the two elements will be connected in series and parallel. The capacitor initially has a charge Q = C VC = (125 F)(100 V) = 12500 C. KVR: Q/C + L dI/dt = 0. Using I = - dQ/dt, A simple harmonic oscillation at  = (LC)- ½.

Q(t) = Q_i cos[t + ]; I(t) = -  Q_i sin[t + ]

The current through an inductor is a continuous function of time. It is zero just before the switch is closed so it is zero just after it is closed. I(t) = -  Q_i sin[] = 0   = 0 and Q_i is just Q(t = 0) or 12500 F. The frequency  is (LC)^{- ½} = (125 x 0.2 x 10^-6 )^{- ½} = (25 x 10^-6 )^{- ½} = 200 ^rad/_s.

I(t) = -  Qi sin[t + ]  Imax =  Qi = (12500 C) (200 rad/s) = 2,500,000 A. B +++ C 100 V +Q 0.2 H -Q - - - Switch closed at t = 0.

The potential drops as one moves down in both elements. Note the pattern in which the static charges accumulate on the inductor to cause the dI/dt . Starting at the lower left and proceeding CW, the sum of the potential changes is Q/C – (L dI/dt) = 0.When the current is in the direction chosen as positive for dI/dt, the charge Q decreases so I = - dQ/dt. Better: Recall that an LC circuit is an oscillator.

Kirchhoff’s law states that the sum of the potential drops is zero. The potential drop between A and B is . The electric field in the integral is the electrostatic field only! A terminal to terminal integration path can be adopted to avoid the regions with circulating field contributions.

The energy initially stored initially in the capacitor is all stored in the inductor when the current is a maximum. ½ L I_max² = Q_max²/2C. I_max = LC]^-½ Q_max =(200 s^-1) (1.25 x 10^-2 C) = 2.50 A

Put the sub-assembly in a circuit and use Kirchhoff’s Laws.

equation	integral form	differential form
Gauss’s Law
Faraday’s Law
Gauss’s Law - Magnetism
Ampere (Maxwell's 4th)
Charge Conservation (Continuity)
Lorentz Force Law

Maxwell’s Equations Explicated

The Helmholtz theorem of vector calculus states that vector fields have two basic behaviors, diverging and circulating. Maxwell has four equations.



The flux of a field through a surface is and in, is the net flux out of the field out of a closed surface. In the field line picture, this is the number of field lines that start in the enclosed volume. The net flux out of a small (infinitesimal) volume divided by the volume is the divergence of the field.

Divergence = Flux out density

The first of Maxwell’s equations states that electric charge density is a divergence source for the electric field. Equivalently, electric charge starts electric field lines. The notation V directs that the flux is to be that through the surface V that bounds the volume V.

The circulation integral of a field is , the integral of the tangential component with respect to path length around a closed path. The component of the curl of the field in the (RHR) normal direction to the area enclosed by the path is the circulation divided by the area enclosed by the path for a small path.

curl = ^circulation/_area or circulation density

The second of Maxwell’s equations states that the electric field circulates in a region of space in which the magnetic field varies in time. This equation represents the induction part of the Faraday’s flux rule. The notation A directs that the circulation path is the path (curve) A that bounds the surface patch A.

The third of Maxwell’s equation states that lines of the magnetic lines do not start or stop. (They may close on themselves.) Equivalently there is no such thing as magnetic charge. The magnetic field does not have a divergence source.

The fourth of Maxwell’s equation contains the fact that the magnetic field circulates around its source which is moving charge. .

The full equation states that the magnetic field circulates in a region of space in which the electric field is time dependent. This behavior is the reciprocity action to that described in the second (Faraday’s) law.

In the case of general time dependence with associated charge acceleration and time dependent charge densities, electromagnetic radiation is generated.

charge	electric field
moving charge	magnetic field
accelerated charge	E-M radiation
x y -2Q Q 1 m 1 m		Ex. 10) A charge of +Q is placed on the x-axis at x =   1 meters, and a charge of -2Q is placed at x = + 1 meters, as shown to the left. At what position on the x-axis will a test charge of +q experience zero net force? (A) (B) – 1/3 m (C) 0 m (D) 1/3 m (E)

One should think about a problem before launching into a solution. The net force on the test charge due to the two existing charges is to be zero. Two vectors can sum to zero only if they are anti-parallel. A little reflection makes it clear that the solution points must lie on the (extended) line joining the two charges. That is: The solution point is somewhere on the x axis.

I

II

This equation has a root in region I and a root in region II. Our previous arguments have identified region I (x < - 1 m) as only region in which both the equal magnitude and opposite direction requirements can be met. Recall that quadratic equations often return one physical root (answer) and one not-so-physical root.

You need x < -1 m; only A is possible.

Gauss:

For applications to conductors, make a sketch. (For equilibrium,) The net charge is on the surface(s). The field at points in the conducting material is zero. A small patch on the surface is essentially planar, and the field only pokes outside. Q_enc = _local A; = A;  away.

Ampere:

 directed to circulate (RHR)

The encircling path through the point where the field is to be determined is to be made of portions perpendicular to the field and parallel to it. : the net parallel length along which the field has a constant non-zero magnitude. _{long st. wire} = 2  r ; _solenoid = ; _toroid = 2  r

Applied to arbitrary paths: Note the direction of each integration path. Give the value of for each path illustrated. CCW: out is positive; CW: in is positive C1: CCW; C2: CW; C3: CW = o Ienc = o Ienc = o [5 A - 2 A] = m = o [- 1 A - 5 A] = - 7.54 T m = o [- 1 A + 2 A] = 1.26 T m

The notation C = A means that C is the curve (path) than bounds the area A. The RHR subscript means that the normal to the area is in the direction dictated by the RHR. In the cases above, an arrow indicating a CCW path means that the normal and hence the positive sense for currents is out of the page. For CW paths, the positive direction is into the page.

Knowledge points:

(a) Field and potential of a point charge: ;

(b) The field of a uniform spherical shell of charge is zero inside and the same as a point charge with q_total located at the center outside.

(c) The field of a spherical ball of charge increases linearly with r inside and is the same as a point charge with q_total located at the center for points outside.

(d) The field of a long uniform line of charge is the charge per length divided by 2 times the distance from the wire. The field is directed away from a uniform line of positive charge.

(e) much, much more …
Ex. 11) A uniform insulating sphere of radius R has a total charge Q and a uniform charge density. The electric field at a point ^R/₃ from the center of the sphere is given by which of the following?

(A) (B) (C) (D) (E)