Answer any three questions: a) Describe the mechanism of current flow in a conductor and derive a relation between current density and drift velocity of electrons. b) What is Seebeck effect? Explain…

a) Mechanism of Current Flow in a Conductor and Relation Between Current Density and Drift Velocity

Mechanism of Current Flow in a Conductor

In a conductor, electric current flows due to the movement of free electrons under the influence of an electric field. The mechanism can be explained as follows:

1. Free Electrons in a Conductor

   • In metals (e.g., copper, aluminum), the outermost electrons of atoms are loosely bound and move freely within the lattice structure.
   • These free electrons are responsible for conducting electricity.

2. Application of Electric Field

   • When a potential difference (voltage) is applied across a conductor, an electric field $\vec{E}$ is established inside it.
   • This electric field exerts a force $\vec{F}=-e\vec{E}$ (where $e$ is the charge of an electron) on the free electrons, causing them to drift in the direction opposite to the electric field (since electrons are negatively charged).

3. Random Thermal Motion vs. Drift Velocity

   • Even without an electric field, free electrons move randomly due to thermal energy (Brownian motion).
   • When an electric field is applied, electrons acquire a net drift velocity $v_d$ in the direction opposite to the electric field.
   • The drift velocity is very small (typically $10^{-4}\text{m/s}$) compared to the random thermal velocities ($10^6\text{m/s}$).

4. Collision and Relaxation Time

   • Electrons frequently collide with the lattice ions (due to thermal vibrations), losing some energy and changing direction.
   • The average time between collisions is called the relaxation time ($\tau$).
   • The drift velocity depends on the electric field and relaxation time: $$v_d=\frac{eE\tau }{m}$$ where:
     • $e$ = charge of an electron ($1.6\times 10^{-19}\text{C}$)
     • $E$ = electric field strength ($\text{V/m}$)
     • $\tau$ = relaxation time ($\text{s}$)
     • $m$ = mass of an electron ($9.11\times 10^{-31}\text{kg}$)

5. Current as Flow of Charge

   • The net current $I$ is the rate of flow of charge through a cross-section of the conductor.
   • If $n$ is the number of free electrons per unit volume (electron density), then the total charge passing through a cross-section per second is: $$I=neA{v}_d$$ where:
     • $A$ = cross-sectional area of the conductor
     • $v_d$ = drift velocity

6. Current Density ($J$)

   • Current density is the current per unit cross-sectional area: $$J=\frac{I}{A}=ne{v}_d$$
   • Substituting $v_d=\frac{eE\tau }{m}$: $$J=ne\left(\frac{eE\tau }{m}\right)=\frac{n{e}^2\tau }{m}E$$
   • The quantity $\frac{n{e}^2\tau }{m}$ is called the conductivity ($\sigma$) of the material: $$J=\sigma E$$
   • This is Ohm’s law in microscopic form.

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Relation Between Current Density and Drift Velocity

From the above derivation, we have: $$J=ne{v}_d$$ Rearranging to express drift velocity: $$v_d=\frac{J}{ne}$$ This shows that:

• Current density ($J$) is directly proportional to drift velocity ($v_d$).
• If the current density increases (due to higher current or smaller cross-sectional area), the drift velocity also increases.
• The drift velocity depends on:
  • Current density ($J$)
  • Electron density ($n$)
  • Charge of electron ($e$)

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b) Seebeck Effect and Variation of Thermo-emf with Temperature

Seebeck Effect

The Seebeck effect is a phenomenon in which a voltage (called thermo-emf or Seebeck emf) is generated in a loop of two different conductors (or semiconductors) when their junctions are at different temperatures.

• Key Observations:

  • If two different metals (e.g., copper and iron) are joined to form a loop, and one junction is heated while the other is kept cold, a voltage difference is observed between the two junctions.
  • This voltage can drive a current in the circuit if it is closed.
  • The effect is reversible: if the temperature difference is reversed, the polarity of the emf also reverses.

• Applications:

  • Thermocouples (used in temperature measurement).
  • Thermoelectric generators (converting heat directly into electricity).

Variation of Thermo-emf with Temperature

When the temperature of the hot junction is gradually increased while keeping the cold junction at $0^{\circ }\mathrm{C}$, the thermo-emf ($\mathcal{E}$) varies as follows:

1. Linear Variation (Approximate)

   • For small temperature differences, the thermo-emf is approximately linear with temperature: $$\mathcal{E}=\alpha \Delta T$$ where:
     • $\alpha$ = Seebeck coefficient (depends on the pair of metals, in $\mu \mathrm{V}{/}^{\circ }\mathrm{C}$)
     • $\Delta T$ = Temperature difference between hot and cold junctions

2. Non-linear Variation (Exact)

   • For larger temperature differences, the relationship is non-linear and can be expressed as: $$\mathcal{E}={\int }_{T_0}^T({\alpha }_A-{\alpha }_B)dT$$ where:
     • ${\alpha }_A$ and ${\alpha }_B$ are the Seebeck coefficients of the two metals.
     • $T_0=0^{\circ }\mathrm{C}$ (cold junction)
     • $T$ = Temperature of the hot junction

3. Graphical Representation

   • If we plot thermo-emf ($\mathcal{E}$) vs. temperature ($T$) of the hot junction, the curve is not a straight line but slightly parabolic or exponential for large $\Delta T$.
   • Example for a copper-iron thermocouple:
     • At low temperatures ($\Delta T<100^{\circ }\mathrm{C}$), $\mathcal{E}\approx 50\mu \mathrm{V}{/}^{\circ }\mathrm{C}\times \Delta T$.
     • At higher temperatures, the slope changes due to temperature dependence of $\alpha$.

4. Neutral Temperature and Temperature of Inversion

   • Neutral Temperature ($T_n$): The temperature at which the thermo-emf is maximum (for some thermocouples).
   • Temperature of Inversion ($T_i$): Beyond this temperature, increasing the hot junction temperature decreases the thermo-emf.

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c) Biot-Savart Law and Magnetic Field Due to a Current-Carrying Circular Coil

Biot-Savart Law

The Biot-Savart Law gives the magnetic field $d\vec{B}$ produced by a small current element $Id\vec{l}$ at a point in space. Mathematically: $$d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{l}\times \hat{r}}{r^2}$$ where:

• ${\mu }_0=4\pi \times 10^{-7}\mathrm{T}\cdot m/A$ (permeability of free space)
• $I$ = current flowing in the wire
• $d\vec{l}$ = small length element of the wire
• $\hat{r}$ = unit vector pointing from the current element to the point where the field is calculated
• $r$ = distance from the current element to the point

Magnetic Field at the Axis of a Circular Coil

Consider a circular coil of radius $R$ carrying current $I$. We want to find the magnetic field at a point $P$ on the axis of the coil, at a distance $x$ from its center.

1. Setup

   • Let the coil lie in the $xy$-plane, centered at the origin.
   • The point $P$ is at $(0,0,x)$ on the $z$-axis.
   • Consider a small current element $Id\vec{l}$ on the coil.

2. Applying Biot-Savart Law

   • The distance from $d\vec{l}$ to $P$ is: $$r=\sqrt{R^2+x^2}$$
   • The direction of $d\vec{l}\times \hat{r}$ is along the $z$-axis (by right-hand rule).
   • The magnitude of $d\vec{B}$ is: $$dB=\frac{{\mu }_0}{4\pi }\frac{Idl\sin (90^{\circ })}{r^2}=\frac{{\mu }_0}{4\pi }\frac{Idl}{r^2}$$ (since $d\vec{l}\perp \hat{r}$, $\sin (90^{\circ })=1$)

3. Component Along the Axis

   • The $z$-component of $d\vec{B}$ is: $$d{B}_z=dB\cos \theta =\frac{{\mu }_0}{4\pi }\frac{Idl}{r^2}\cos \theta$$ where $\theta$ is the angle between $d\vec{l}$ and $\hat{r}$.
   • From geometry, $\cos \theta =\frac{x}{r}$, so: $$d{B}_z=\frac{{\mu }_0}{4\pi }\frac{Idlx}{r^3}$$

4. Integrating Over the Entire Coil

   • The total field $B$ is the integral of $d{B}_z$ over the entire loop: $$B=\int d{B}_z=\frac{{\mu }_{0}Ix}{4\pi r^3}\int dl$$
   • Since $\int dl=2\pi R$ (circumference of the coil): $$B=\frac{{\mu }_{0}Ix}{4\pi r^3}\times 2\pi R=\frac{{\mu }_{0}IRx}{2r^3}$$
   • Substituting $r=\sqrt{R^2+x^2}$: $$B=\frac{{\mu }_{0}IRx}{2(R^2+x^2{)}^{3/2}}$$

5. Special Cases

   • At the center of the coil ($x=0$): $$B=0$$ (The field cancels out due to symmetry.)
   • Far from the coil ($x\gg R$): $$B\approx \frac{{\mu }_{0}IRx}{2x^3}=\frac{{\mu }_{0}IR}{2x^2}$$ (The coil behaves like a magnetic dipole.)

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d) Faraday’s Law of Electromagnetic Induction and Expression for Induced emf in a Rotating Coil

Faraday’s Law of Electromagnetic Induction

Faraday’s law states that:

"The induced electromotive force (emf) in a closed loop is equal to the negative rate of change of magnetic flux through the loop."

Mathematically: $$\mathcal{E}=-\frac{d{\Phi }_B}{dt}$$ where:

• $\mathcal{E}$ = induced emf (in volts)
• ${\Phi }_B$ = magnetic flux through the loop (in webers, $\mathrm{Wb}$)
• ${\Phi }_B=\int \vec{B}\cdot d\vec{A}$

Lenz’s Law (Extension of Faraday’s Law)

Lenz’s law adds that:

"The direction of the induced emf (and current) is such that it opposes the change in magnetic flux that produced it."

Expression for Induced emf in a Rotating Rectangular Coil

Consider a rectangular coil of area $A$ and $N$ turns, rotating with angular velocity $\omega$ in a uniform magnetic field $B$.

1. Magnetic Flux Through the Coil

   • The magnetic flux through one turn is: $${\Phi }_B=BA\cos \theta$$ where $\theta =\omega t$ (angle between $\vec{B}$ and the normal to the coil).
   • For $N$ turns: $${\Phi }_B=NBA\cos \omega t$$

2. Applying Faraday’s Law

   • The induced emf is: $$\mathcal{E}=-\frac{d{\Phi }_B}{dt}=-\frac{d}{dt}(NBA\cos \omega t)$$
   • Differentiating: $$\mathcal{E}=NBA\omega \sin \omega t$$
   • The maximum emf occurs when $\sin \omega t=1$: $${\mathcal{E}}_{\text{max}}=NBA\omega$$

3. Instantaneous emf

   • The instantaneous emf is: $$\mathcal{E}(t)=NBA\omega \sin \omega t$$
   • This is an AC emf with amplitude $NBA\omega$.

4. Average emf Over One Cycle

   • The average value of $\sin \omega t$ over a full cycle is zero, but the root mean square (rms) emf is: $${\mathcal{E}}_{\text{rms}}=\frac{{\mathcal{E}}_{\text{max}}}{\sqrt{2}}=\frac{NBA\omega }{\sqrt{2}}$$

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