Current Density Vector

Q. A copper wire of length $10m$ and radius $\left({10}^{-2}/\sqrt{\pi }\right)m$ has electrical resistance of $10\Omega$. The current density in the wire for an electric field strength of $10\left(V/m\right)$ is:

1. ${10}^{6}A/m^2$
2. ${10}^{5}A/m^2$
3. ${10}^{-5}A/m^2$
4. ${10}^{4}A/m^2$

Q. (a) Define the term 'conductivity' of a metallic wire. Write its SI unit.
(b) Using the concept of free electrons in a conductor, derive the expression for the conductivity of a wire in terms of number density and relaxation time. Hence obtain the relation between current density and the applied electric field E.

Q. The current density across a cylindrical conductor of radius $R$ varies according to the equation $J=J_0(1-\frac{r}{R}),$ where $r$ is the distance from the axis. Thus, the current density is a maximum $J_0$ at the axis $r=0$ and decreases linearly to zero at the surface $r=R$. Calculate the current, $I$ in terms of $J_0$ if the conductor's cross sectional area is $A=\pi R^2$.

1. $I=\frac{J_{0}A}{3}$
2. $I=\frac{J_{0}A}{6}$
3. $I=\frac{J_{0}A}{9}$
4. $I=\frac{J_{0}A}{12}$

Q.

An electron of mass $m$ and magnitude of charge$\left|e\right|$ initially at rest gets accelerated by a constant electric field $E$. The rate of change of de-Broglie wavelength of this electron at time $t$ ignoring relativistic effects is

1. $-\frac{\left|e\right|Et}{h}$

2. $-\frac{h}{\left|e\right|E\sqrt{t}}$

3. $-\frac{h}{\left|e\right|E{t}^2}$

4. $-\frac{2h{t}^2}{\left|e\right|E}$

Q. A current of $5\text{A}$ is passing through a non-linear magnesium wire of cross-section $0.04{\text{m}}^2$. At every point, the direction of current density is at an angle of ${60}^{\circ }$, with the unit vector of area of cross-section. The magnitude of electric field at every point of the conductor is :

Resistivity of magnesium is $44\times {10}^{–8}\Omega \text{-m}$.

1. $11\times {10}^{-7}\text{V/m}$
2. $11\times {10}^{-3}\text{V/m}$
3. $11\times {10}^{-5}\text{V/m}$
4. $11\times {10}^{-2}\text{V/m}$

Q.

A potential difference $\mathrm{V}$ is applied across a conductor of length $‘\mathrm{l}’$. How is the drift velocity affected when $\mathrm{V}$ is doubled and $‘\mathrm{l}’$ is halved?

Q.

Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are $M{L}^2{I}^{-2}{T}^{-3}$ and $M{L}^2{T}^{-3}{I}^{-1}$ respectively.

Q. The current density across a cylindrical conductor of radius $R$ varies in magnitude according to the equation $J=J_0(1-\frac{r}{R})$ where $r$ is the distance from the central axis. Thus, the current density is maximum $J_0$ at the axis ($r=0$) and decreases linearly to zero at the surface $(r=R)$. The current in terms of $J_0$ and conductor's cross-sectional area $A$ is :-

1. $\frac{J_{0}A}{3}$
2. $\frac{J_{0}A}{6}$
3. $\frac{J_{0}A}{2}$
4. $\frac{J_{0}A}{5}$

Q. A non uniform electric field is given by the expression \overrightarrow E =ay\overrightarrow i +bz \overrightarrow j + cx\overrightarrow{} , where a , b , c are cons†an ts . Determine the electric flux through a rec†an gular surface in the xy plane , extending from x=0 to x= w and from y=0 to y=h

Q. An electric current of 8 Amperes is measured across an area of $50c{m}^2$, then current density in this area is:

1. Must be greater than or equal to $1600A/m^2$
2. Must be less than or equal to $1600A/m^2$
3. Cannot say
4. Must be $1600A/m^2$

Q. The variation of current and voltage in a conductor has been shown in figure. The conductance of the conductor is (Give your answer in ${\Omega }^{-1}$).

1. $\frac{1}{2}$
2. $\frac{1}{5}$
3. $\frac{1}{3}$
4. $\frac{1}{4}$

Q. A copper wire of length 1m and radius 1mm is joined in series with an iron wire of length 2m and radius 3mm and a current is passed through the wire. The ratio of current densities in the copper and iron wire is

1. 18 : 1
2. 9 : 1
3. 6 : 1
4. 2 : 3

Q.

Does de-Broglie wavelength depend on charge?

Q. A Conductor with rectangular cross-section has dimensions $(a\times 2a\times 4a)$ as shown in figure. Resistance across $\text{AB}$ is $x,$ across $\text{CD}$ is $y$ and across $\text{EF}$ is $z.$ Then

1. $x=y=z$
2. $x>y>z$
3. $y>z>x$
4. $x>z>y$

Q.

why is it so that when the diametre of a conductor is doubled, then drift velocity of electrons inside it will not change

Q. The electric current in an $\text{X}$- ray tube (from the target to the filament) operating at $40\text{kV}$ is $10\text{mA}$. Assume that on an average, $1\%$ of the total kinetic energy of the electrons hitting the target are converted into $X-\text{rays}$. The total power $\left(\text{in W}\right)$ emitted as $\text{X}$-rays is (Integer only)

Q.

A $150m$ long metal wire connects points $A$ and $B$. The electric potential at point $B$ is $50V$ less than that at point $A$. If the conductivity of the metal is $60\times {10}^{6}mho/m$ then the magnitude of the current density in the wire is equal to?

Q.

A long straight metal rod has a very long hole of radius ′a′ drilled parallel to the rod axis as shown in the figure. If the rod carries a current ′I′. Find the magnetic induction on the axis of the hole, where$\mathrm{OC}=\mathrm{c}$:

1. $\frac{{\mathrm{\mu }}_0\mathrm{Ic}}{\mathrm{\pi }\left({\mathrm{b}}^2-{\mathrm{c}}^2\right)}$

2. $\frac{{\mathrm{\mu }}_0\mathrm{Ic}}{2\mathrm{\pi }\left({\mathrm{b}}^2-{\mathrm{c}}^2\right)}$

3. $\frac{{\mathrm{\mu }}_0\mathrm{I}\left({\mathrm{b}}^2-{\mathrm{c}}^2\right)}{2\mathrm{\pi c}}$

4. $\frac{{\mathrm{\mu }}_0\mathrm{Ic}}{2{\mathrm{\pi a}}^2{\mathrm{b}}^2}$

Q. A long straight cylindrical region of radius $a$ carries a current along its length. The current density $\left(J\right)$ varies from the axis to the edge of the cylindrical region according to $J=J_0(1-\frac{r}{a})$ , where $r$ is distance from the axis $(0\le r\le a)$. Choose the correct answer (s) among the following.

1. Mean density $\overline{J}=\frac{J_0}{3}$
2. At $r=0;B=0$
3. At $r=a;B=\frac{{\mu }_0{J}_{0}a}{6}$
4. The $B-r$ graph is parabolic with maxima at distance $r=\frac{a}{2}$

Q. Find the current flowing through a copper wire of length $0.2\text{m}$, area of cross-section $1{\text{mm}}^2$, when connected to a battery of $4\text{V}$. Given that electron mobility $=4.5\times {10}^{-6}\frac{{\text{m}}^2}{\text{Vs}}$ and charge on electron $=1.6\times {10}^{-19}\text{C}.$ The number density of electron in copper is $8.5\times {10}^{28}{\text{m}}^{-3}$.

1. $1.22\text{A}$
2. $2.12\text{A}$
3. $4.52\text{A}$
4. $4.68\text{A}$

Q.

A wire of ${10}^{-3}c{m}^2$ cross sectional area carries a current of 0.25 A, then current density is

1. 1.5×10⁶ Am^-2

2. 2.5×10⁶ Am^-2

3. 3.5×10⁶ Am^-2

4. 4.5×10⁶ Am^-2

Q. { †ext { An electric dipole of length } 10 cm †ext { having charges } + 6 × 10 ^ { - 3 } C †ext { and } - 6 × 10 ^ { - 3 } C †ext { , placed at } } { 30 ^ { ° } †ext { with respect to a uniform electric field experiences a torque of magnitude } 6 \sqrt { 3 } Nm †ext { . } } { †ext { Calculate the magnitude of electric field. }

Q. For a metallic conductor, what is the relation between current density $\left(j\right)$, conductivity $\left(\sigma \right)$ and electric field Intensity $E$?

Q.

When will the conductivity become equal to the molar conductivity?

Q. At some higher altitude, density of free electrons is around
${10}^{12}$ per cubic meter and due to low density of air the mean path of an electron is about 0.1m. If the mean speed of electrons is ${10}^{5}m/s$, then the conductivity of atmosphere is
(answer upto 2 decimal points)

Q. I apply an electric field of ${10}^{5}N/C$ on a conductor of cross section $4m{m}^2$ and I get a current of 1A. If I want to change the current to 0.3A, what is the electric field I need to apply?

1. $5\times {10}^{4}N/C$
2. $4\times {10}^{4}N/C$
3. $3\times {10}^{4}N/C$
4. $2\times {10}^{4}N/C$

Q. The current density across a cylindrical conductor of radius $R$ varies according to the equation $J=J_0(1-\frac{r}{R}),$ where $r$ is the distance from the axis. Thus, the current density is a maximum $J_0$ at the axis $r=0$ and decreases linearly to zero at the surface $r=R$. Calculate the current, $I$ in terms of $J_0$ if the conductor's cross sectional area is $A=\pi R^2$.

1. $I=\frac{J_{0}A}{12}$
2. $I=\frac{J_{0}A}{9}$
3. $I=\frac{J_{0}A}{6}$
4. $I=\frac{J_{0}A}{3}$

Q. Explain the term 'drift velocity' of electrons in a conductor. Hence obtain the expression for the current through a conductor in terms of 'drift velocity'.

Q. A wire of resistance $12\Omega /\text{m}$ is bent to form a complete circle of radius $10\text{cm}$. The resistance between its two diametrically opposite points, $A$ and $B$ as shown in the figure is -

1. $6\Omega$
2. $0.6\pi \Omega$
3. $3\Omega$
4. $6\pi \Omega$

Q. The electric field intensity $E,$ current density $J$ and specific resistance $k$ are related to each other through the relation

1. $E=\frac{J}{k}$
2. $E=Jk$
3. $E=\frac{k}{J}$
4. $k=JE$