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A gold wire of length $13.7\ \mathrm{cm}$ and diameter $0.188\ \mathrm{mm}$ is going to be used to construct a thermocouple. Gold has a temperature coefficient of resistivity of $\alpha =3.4\times {10}^{-3}{}^\circ {\rm C}^{-1}$.

(a) At $20{}^\circ {\rm C}$, gold has a resistivity of ${\rho }_{Au}=2.44\times {10}^{-8}\, \mathrm{\Omega }\mathrm{.m}$, what is the resistance of this wire at this temperature?

(b) After constructing out thermocouple, we place it into a liquid of unknown temperature and apply a voltage across the leads of the thermocouple of $0.224\ \mathrm{V}$. If we measure a current of $0.884\ \mathrm{A}$ going through this thermocouple, what is the temperature of this liquid in ${}^\circ {\rm C}$?

A cylindrical wire conductor of radius $\mathrm{a}$ as shown in the figure has a resistivity $\rho $ and carries a constant current .

(a) What is the magnitude and direction of the electric field vector $\vec{E}$ at the point just inside the wire at a distance a from the axis. Hint: recall that $\vec{J}=\sigma \vec{E}$ and that $I=JA$ where $A=\pi a^2$ is the cross-sectional area of the wire.

(b) What is the magnitude and direction of the magnetic field $\vec{B}$ just inside the wire at a distance $a$ from the axis?

(c) What is the magnitude and the direction of the Poynting vector $\vec{S}=\left(\frac{1}{{\mu }_0}\right)\left(\vec{E}\times \vec{B}\right)$ at this same point? The magnitude and direction of $\vec{S}$ is the rate and direction at which electromagnetic energy ($W/m^2$) is flowing into or out of the conductor.

(d) Recalling that $\vec{S}$ is the magnitude and direction of electromagnetic energy flow into or out of a conductor and use your part (c) answer to determine the rate of energy flow into the volume occupied by a length $l$ of the conducting wire. Hint: integrate $\vec{S}$ over the surface of this volume and compare your result to the rate of heat generation ($RI^2$ losses) in the same volume of wire.

The resistivity of gold is $2.44\times {10}^{-8}\mathrm{\Omega }\mathrm{.m}$ at room temperature. A gold wire that is $0.9\ \mathrm{mm}$ in diameter and $14\ \mathrm{cm}$ long carries a current of $940\ \mathrm{mA}$. What is the electric field in the wire?

Thirteen resistors are connected across point A and B as shown in the figure. If all the resistors are accurate to 2 significant figures, what is the equivalent resistance between points A and B?

The figure shows a $2.0\ \mathrm{cm}$ diameter roller that turns at $90\ \mathrm{rpm}$. A $4.0\ \mathrm{cm}$ wide plastic film is being wrapped onto the roller, and this plastic carries an excess electric charge having a uniform surface charge density of $5.0\ \mathrm{nC}/{\mathrm{cm}}^{\mathrm{2}}$. What is the current of the moving film?

A $1\,\mu\mathrm{F}$ capacitor has a potential difference of $6.0\ \mathrm{V}$ applied across its plates. If the potential difference across its plates is increased to $8.0\ \mathrm{V}$, how much __additional__ energy does the capacitor store?

A parallel plate capacitor consists of two square plates of side length $10\, \mathrm{cm}$ that are spaced $5\, \mathrm{mm}$ apart with air between the plates. The plates are connected to a $200\,\mathrm{V}$ battery.

(a) Calculate the capacitance.

(b) Calculate the electric field between the plated.

(c) Calculate the amount of charge stored on each plate.

(d) Calculate the potential energy stored in the capacitor.

(e) An electron is released from the negative plane with a speed of $1.5\times {10}^7\,\mathrm{m/s}$ and accelerates towards the positive plate. What is the electron's speed when it strikes the positive plate?

(f) During this process, what is the electron's change in kinetic energy in electron volt?

A nichrome wire is $50\ \mathrm{cm}$ long. The resistivity of the nichrome is $1.5\times {10}^{-6}\mathrm{\Omega }.\mathrm{m}$. The potential difference between the ends is $4.0\,\mathrm{V}$ and a current of $2.0\ \mathrm{A}$ is flowing through the wire.

(a) Calculate the resistance of the wire

(b) Calculate the diameter of the wire's circular cross section.

(c) Calculate the power dissipated in the wire.

(d) How many electrons per second pass through the cross section of the wire?

Two conductor made of the same material are connected across the same potential difference. Conductor A has twice the diameter and twice the length of conductor B. What is the ratio of the power delivered to A to the power delivered to B?

A laser pulse that lasts $12\ \mathrm{ns}$ and delivers $255\,\mathrm{kW}$ of power to a spot size $0.15\mathrm{cm}$ on a target.

(a) What is the total energy delivered per pulse?

(b) If the wavelength of the laser $632\,\mathrm{nm}$, what is the frequency of the $EM$ wave?

A parallel-plate capacitor is constructed using two conducting plates of area $A$. They are placed at a separation of distance $d$ and initially there is vacuum between the plates. The capacitor is connected to a power source and is kept at a potential of $V$. A dielectric slab of area $A$ and thickness $d/2$ and dielectric constant $\kappa $ is inserted between the two plates.

(a) Calculate the charge on the plates before the slab is inserted.

(b) Calculate the charge on the plates after the slab is inserted.

(c) Calculate the work done in inserting the dielectric.

A dielectric slab of thickness $d$ and dielectric constant $K$ is inserted inside a capacitor with area $A$ and thickness $L$. A voltage difference $V$ is applied across the plats.

(a) Find the electric fields inside the air gap and inside the dielectric

(b) Find the surface charge density on each of the plates.

(c) Use the information in (b) to obtain the capacitance.

(d) Find the capacitance of the system using capacitors connected in series, and compare your result with part (c).

A parallel plate capacitor with plates of area $A$ is filled with two dielectrics (of dielectric constants ${\kappa }_1$ and ${\kappa }_2$. Each dielectric slab has thickness $d/2$. Find the capacitance of the two arrangements in figures.

Four capacitors, each with a capacitance $10\,\mu\mathrm{F}$ are connected as shown in the figure to a $10\, \mathrm{V}$ battery. What is the charge on the capacitor $C_1$?

A cylindrical wire of radius $R$ carries a nonuniform current density $J(r)\ =\ ar$, where $r$ is the distance from the axis of the cylinder and $a$ is a constant. What is the total current $i$ which flows in the wire?

Consider the circuit shown at the right, where all resistors have the same resistance $R$. At $t=0$ with the capacitor $C$ uncharged, the switch is closed.

(a) At $t=0$ the three currents can be determined by analyzing a simpler, but equivalent, circuit. Identify this simpler circuit and use it to find the values of $I_1$, $I_2$ , and $I_3$ at $t=0$.

(b) At $t=\infty $, the currents can be determined by analyzing another equivalent circuit. Identify this simpler circuit and use it in finding the values of $I_1,\ I_2$ and $I_3$ at $t=\infty $.

(c) At $t=\infty $, what is the potential difference across the capacitor?

In the figure at the right assume $\mathrm{R\ =\ 1.2\ k}\mathrm{\Omega }$. What is the equivalent resistance of the circuit connected to the battery?

(a) A dielectric capacitor with a capacitance of $C_0=20\,\mu{\rm F}$ is charged up to $V_0=60\, {\rm V}$. What is the energy $U_0$ stored in the capacitor?

(b) The capacitor is then disconnected from everything else. The dielectric is removed, changing the capacitance to $C_1=10\,\mu {\rm F}$. What was the dielectric constant $\kappa$ of the dielectric?

(c) The capacitor is still disconnected. What are the new charge $Q_1$, voltage $V_1$, and energy $U_1$ stored in the capacitor?

(d) Did the energy of the capacitor increase or decrease? Where did the energy come from/go to?

The capacitance of a cylindrical capacitor can be increased by:

(a) decreasing both the radius of the inner cylinder and the length

(b) increasing both the radius of the inner cylinder and the length

(c) increasing the radius of the outer cylindrical shell and decreasing the length

(d) decreasing the radius of the inner cylinder and increasing the radius of the outer cylindrical shell

(e) only by decreasing the length

The capacitance of a cylindrical capacitor, comprising of a long cylindrical conductor with

radius $r_a$ and linear charge density $+\lambda $, and a coaxial cylindrical conducting shell with radius $r_b$ , $r_b>r_a$, and linear charge density $-\lambda $ is calculated as

\[C=\frac{2\pi {\epsilon }_0L}{{\mathrm{ln} r_b\ }-{\mathrm{ln} r_a\ }}=\frac{2\pi {\epsilon }_0L}{{\mathrm{ln} \left(\frac{r_b}{r_a}\right)\ }}\]

Where $L$ is the length of the cylinder. As you can see, the capacitance is inversely proportional to the difference of logarithm of the inner and outer cylinder. Therefore by increasing the inner radius $r_a$, its logarithms is also increasing. As a result, difference between the two logarithms is decreasing and consequently the capacitance of the system is increased.

The correct answer is B.

A $20\ \mathrm{\mu }\mathrm{F}$ capacitor is charged to $200\ \mathrm{V}$. its stored energy is:

(a) $4000\ \mathrm{J}$

(b) $4\ \mathrm{J}$

(c) $0.4\ \mathrm{J}$

(d) $0.1\ \mathrm{J}$

(e) $0.004\ \mathrm{J}$

Which of the following graphs best represents the current-voltage relationship for a device that obeys Ohm's law?

Category : Capacitance and Resistance

MOST USEFUL FORMULA IN CAPACITANCE AND RESISTANCE:

Capacitor: any pair of conductors separated by an insulating material.

\[C=\frac Q V\]

The SI unit of capacitance is Farad

\[\mathrm {1\,F=1\, C/V}\]

$Q$ is the conductor's total charge and $V$ is the potential drop across the capacitor.

Capacitance of a parallel plate capacitor in vacuum:

\[C=\epsilon_0 \frac A D\]

Capacitance of a spherical capacitor:

\[C=4\pi\epsilon_0 \frac{r_ar_b}{r_b-r_a}\]

where $r_b$ and $r_a$ are the outer and inner radii,respectively.

Capacitance of a spherical capacitor:

\[C=\frac{2\pi\epsilon_0L}{\ln\left(\frac {r_b}{r_a}\right)}\]

where $r_b$ and $r_a$ are the outer and inner radii,respectively.

Capacitors in Series:

\[\frac 1{C_{eq}}=\frac 1{C_1}+\frac 1{C_2}+\frac 1{C_3}+\cdots\]

In a series connection the magnitude of charge on all plates is the same.

Capacitors in parallel:

\[C_{eq}=C_1+C_2+C_3+\cdots\]

In a parallel connection the potential drop across each capacitor is the same.

Potential energy stored in a capacitor:

\[U=\frac{Q^{2}}{2C}=\frac 1 2 CV^2=\frac 1 2 QV\]

Electric energy density in a vacuum:

\[u=\frac 1 2 \epsilon_0 E^2\]

Definition of current:

\[I=\frac{dQ}{dt}\]

Definition of current density:

\[J=\frac I A\]

Definition of resistivity:

\[\rho=\frac E J\]

Resistance of a conductor:

\[R=\rho \frac L A\]

The SI unit of resistance is ohm ($\Omega$).

Ohm's law:

\[I=\frac V R\]

Number Of Questions : 21

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