P142 Test #1Prof. Cline

1230-1400 hours, 14 October 2004

1. THIS IS A CLOSED BOOK EXAM

2. SHOW ALL STEPS TO GET FULL CREDIT.

3. Do all parts of each problem.

4. Making a diagram when appropriate usually aids in solving the problem.

5. Circle your answers and give units when appropriate.

6. Necessary constants and formulae are given on the separate sheet.

1;[25pts] a; Write down Maxwell's equation for the flux of E for electrostatics. Describe the meaning and physical laws underlying this equation.

b: Write down Maxwell's equation for the circulation of E for electrostatics. Describe the meaning and physical laws underlying this equation.

An electric dipole comprises charges $+Q$ and $-Q$ separated by a distance $d.$

c; Derive the electric potential at a distance $r$ where $r>>d$.

d: Derive the $E$ field along the axis of the dipole for $r>>d.$




2;[30 pts]A long coaxial cable comprises two concentric cylindrical conductors, an inner thin copper cylindrical shell of radius $a$, surrounded by a thin copper cylindrical shell of inner radius $b$. The inner conductor has a static linear charge density of $+\lambda $C/m, while the outer shell has a linear charge density of $-\lambda $C/m.

a;Derive the E field inside the inner coaxial electrode.

b;Derive the E field between the two cylindical shells.

c;Derive the electric field outside the outer cylindrical shell.

d;Derive the electric potential difference between the cylinders.

e:Derive the capacitance per meter of cable.

f:Derive the stored energy per meter of cable

g:Describe exactly where the charge density resides on both the inner and outer cylinders, that is, is it distributed throughout the conductor, or concentrated on which surfaces.




3:[20pts]In the circuit shown the batteries have negligible internal resistance.

a: Find the current in each resistor

b: Find the power provided by each battery

c: Find the power dissipated in each resistor.
test1.3.jpg
4:[25pts] a; Write down Maxwell equation for the flux of B for magnetostatics. Describe the meaning and physical laws underlying this equation.

b: Write down Maxwell's equation for the circulation of B for magnetostatics. Describe the meaning and physical laws underlying this equation.

An infinitely long straight solid cylindrical conductor of radius $R$ carries a steady current density $\QTR{bf}{j}\ $uniformly throughout the conductor and into the paper except it is zero inside a cylindrical cavity of radius $a=\frac{R}{4}$ whose axis is parallel to the axis of the conductor and a distance $b=\frac{R}{2}$ from it. Consider this problem as the superposition of a uniform current density thoughout the cylinder of radius $R$ plus an equal and opposite current density in the cylindrical cavity that cancels the current density.

c: Derive the magnitude and direction of the magnetic field at $r=\frac{R}{4} $ on the $x$ axis

d: Derive the magnitude and direction of the magnetic field at $r=\frac{R}{2} $ on the $x$ axis

e: Derive the magnitude and direction of the magnetic field at $r=\frac{3R}{4}$ on the $x$ axis
test1.4.jpg

Constants

MATH MATH

$\mu _0=4\pi $ $10^{-7}N/A^2$

$\Pr $oton ch$\arg $e = - electron charge = $1.60210^{-19}C.$

$1$ $eV$=$1.60210^{-19}J$

$c=2.998$ $10^{8}m/s$

Maxwell's equations for statics

MATH

MATH

MATH

MATH MATH

MATH

Lorentz force

MATH

Electrostatics

Coulomb's Law:
MATH

Superposition:
MATH

Electric field:
MATH

Electric potential:
MATH




MATH

MATH

Capacitance: $Q=CV$

Stored energy: MATH

Electric dipole: MATH

Torque on dipole:
MATH

Electric current

Ohm's law:
MATH $V=IR$

Power: $P=VI$

Kirchhoff's rules, for steady state

Loop rule: Algebraic sum of potential differences

is zero around a closed circuit

Node rule: Sum of currents into a junction is zero

Magnetism

Magnetic field of moving point charge:
MATH

Biot Savart law:
MATH

Force on current loop:
MATH

Magnetic moment of current loop: MATH

Torque on current loop:
MATH

Miscellaneous

Centripetal acceleration
$a=\frac{v^{2}}{r}$

Integrals:


MATH


MATH







1;[25pts] a; Write down Maxwell's equation for the flux of E for electrostatics. Describe the meaning and physical laws underlying this equation.

MATH

This results from the fact that, as given by Coulomb's law, the electric field due to a point charge is radial and falls off as the inverse square of the distance $r.$ It also assumes superposition



b: Write down Maxwell's equation for the circulation of E for electrostatics. Describe the meaning and physical laws underlying this equation.

MATHThis is a statement that for electrostatics the electric field for a point charge is radial. It implies that the static electric field is conservative.

An electric dipole comprises charges $+Q$ and $-Q$ separated by a distance $d.$

c; Derive the electric potential at a distance $r$ where $r>>d$.

Consider the electric dipole shown in the figure


4.4.jpg
Electric field components at P caused by an electric dipole

The electric potential at the point $P$ is: MATHLet $r>>a$ then MATH, and the equation simplifies to: MATH

The electric dipole moment is given by: MATH that is, the electric dipole moment is a vector quantity as opposed to the monopole moment q.

Thus the electric potential can be written as: MATHsince MATH where MATH point from the electric dipole towards the point $P$.

d: Derive the $E$ field along the axis of the dipole for $r>>d.$

The electric field for the electric dipole, in the far-field approximation, can be derived from the above potential since MATH For this case, it is better to express the gradient in spherical rather than cartesian coordinates. Thus: MATHwhich leads to: MATHFor the simple case along the dipole axis, $\theta =0,$ only the radial field is non zero. That is since $\theta =0$ then

MATH




2;[30 pts]A long coaxial cable comprises two concentric cylindrical conductors, an inner thin copper cylindrical shell of radius $a$, surrounded by a thin copper cylindrical shell of inner radius $b$. The inner conductor has a static linear charge density of $+\lambda $C/m, while the outer shell has a linear charge density of $-\lambda $C/m.

a;Derive the E field inside the inner coaxial electrode. $r<a$

Use Gauss' Law plus cylindrical symmetry inside the inner cylinder. For a concentric cylindrical Gaussian surface we get using MATH

Since the enclosed charge is zero inside the inner shell.

b;Derive the E field between the two cylindical shells. $a<r<b$

Using Gauss'Law as above we get MATHradially outwards from the inner cylinder.

c;Derive the electric field outside the outer cylindrical shell $r>b$

Use Gauss' Law plus cylindrical symmetry outside the outer cylinder. For a concentric cylindrical Gaussian surface we get using MATHsince the net charge enclosed is zero.

d;Derive the electric potential difference between the cylinders.

MATH

e:Derive the capacitance per meter of cable.

The capacitance is just $C=\frac{Q}{V}$ thus we have the capacitance per meter of length as MATH

f:Derive the stored energy per meter of cable

The stored energy is given by MATHThus the energy stored per meter of cable is MATH

g:Describe exactly where the charge density resides on both the inner and outer cylinders, that is, is it distributed throughout the conductor, or concentrated on which surfaces.

The charge resides on the outside of the inner cylinder and inside of the outer cylinder in order for the electric field to be zero inside the conductors.

3:[20pts]In the circuit shown the batteries have negligible internal resistance.

a: Find the current in each resistor


test1.3.jpg

a: Find the current in each part of the circuit

Assume $I_{1}$ flows clockwise in the through the $1\Omega $ and $2\Omega $ resistors in the top left of the figure. Let $I_{2}$ flow upwards in the center branch while $I_{3}$flows downwards in the $5$ $\Omega $ resistor.

Kirchhoff's Node rule gives that MATH The Loop Rule gives the two equations MATHand MATH

Solving these gives $I_{1}=2A$, $I_{2}=-1.0A$ and MATH

b: Find the power provided by each battery

We know that the power is given by MATHTherefore MATHThat is the right hand $4V$ battery is being charged by the other batteries.

c: Find the power dissipated in each resistor.

The power dissipated in the resistors is given by MATH$.$ Thus the power dissipated in the resistors isMATH

4:[25pts] a; Write down Maxwell equation for the flux of B for magnetostatics. Describe the meaning and physical laws underlying this equation.




[ANS]MATH

This is a statement that implies that the magnetic field due to a line current or moving charge is tangential. There are no magnetic monopoles that can serve as sources or sinks of the magnetic field.

b: Write down Maxwell's equation for the circulation of B for magnetostatics. Describe the meaning and physical laws underlying this equation.




[ANS]MATH

An electrical current produces a tangential magnetic field circulating around the current in a clockwise direction.

An infinitely long straight solid cylindrical conductor of radius $R$ carries a steady current density $\QTR{bf}{j}\ $uniformly throughout the conductor and into the paper except it is zero inside a cylindrical cavity of radius $a=\frac{R}{4}$ whose axis is parallel to the axis of the conductor and a distance $b=\frac{R}{2}$ from it. Consider this problem as the superposition of a uniform current density thoughout the cylinder of radius $R$ plus an equal and opposite current density in the cylindrical cavity that cancels the current density.
test1.4.jpg

c: Derive the magnitude and direction of the magnetic field at $r=\frac{R}{4} $ on the $x$ axis

[ANS]

Consider this problem as the superposition of two equal and opposite cylindrical current densities. The current in the cylinder of radius $R$ is flowing into the paper whjile the current in the cylinder of radius $a$ flowing out of the paper which cancels the current inside this smaller cylinder. Consider the $B$ field inside the cylinder of radius $R$ with uniform current density $j$ into the page




Use Ampere's Law

$\hspace{0.9in}$MATHBy symmetry one must have cylindrical symmetry and the magnetic field must be tangential to ensure that MATHThus assuming the line integral is taken in a right handed direction with respect to $I$ then for a concentric circle we have MATHThat is MATHAt $r=x=\frac{R}{4}$ the $B$ field on the $x$ axis is

MATH

The $B$ field due to the radius $a$ cylinder of current $j$ coming out of the page at $r=\frac{R}{4}$ $\ $ is given by MATHusing the same argument as above.

Superposition of the two current densities produces the desired current density and the net $B$ filed is the sum of the two $B$ fields, i.e.MATH

d: Derive the magnitude and direction of the magnetic field at $r=\frac{R}{2} $ on the $x$ axis

The argument is as above except that $r=\frac{R}{2}$ for the larger cylinder for which the $B$ field is MATHThe $B$ field for the smaller radius cylinder at its center ius zero. Thus supoerposition gives a net $B$ field of MATH

e: Derive the magnitude and direction of the magnetic field at $r=\frac{3R}{4}$ on the $x$ axis




For the larger radius cylinder at $r=\frac{3R}{4}$ the $B$ field is given by MATHThe $B$ field due to the smaller radius cylinder of current is given by MATHNote that these two fields are in opposite directions anc partially cancel when superimposed. That is MATH

Thus we have a situation where along the $x$ axis the field is constant over the diameter of the smaller hole. Note that off the $x$ axis the field is more complicated since then the $B$ fields from the two cylinders are not colinear.