Gaussian Surface and its Properties

Gaussian Surface and its Properties:

The Gaussian surface is a hypothetical or imaginary closed three-dimensional surface. This surface is used to calculate the electric flux through a vector field (i.e. gravitational field, electric field, or magnetic field).

Examples:

Gaussian surfaces are surfaces of spheres, cylinders, cubes, etc. There are some surfaces which cannot be used as Gaussian surfaces like the surface of disc, square etc.

Essential properties of Gaussian surface are :

1. The Gaussian surface must be closed surface to clearly define the regions, inside, on, and outside the surface.

2. A Gaussian surface is constructed to pass through the point at which the electric field is being calculated.

3. The shape of the Gaussian surface depends upon the shape or symmetry of the charge distribution (i.e. the source).

4. For systems with discrete charges, the surface should not intersect any point charge, as the electric field is undefined at the location of a point charge. However, the surface can intersect continuous charge distributions

5. The electric flux through the surface depends solely on the total charge enclosed within it, not on the external charges.

6. The electric field at any point on the Gaussian surface is influenced by both internal and external charges.

7.If the electric flux is zero through the surface, it does not necessarily mean the electric field is zero. However, if the electric field is zero at every point on the surface then the electric flux will be definitely zero.

8. If a closed surface encloses no net charge, the total electric flux through it will be zero—regardless of whether the external electric field is uniform or varying.

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Population of energy level and it thermal equilibrium condition

Population of energy level:

The number of atoms per unit volume in any energy level is called the population of that energy level.

The population $N$ of any energy level $E$ depends on the temperature $T$ which can be described by

$N=e^{-\left(\frac{E}{kT}\right)}$

Where $k \rightarrow$ Boltzmann's Constant

The above equation is called the Boltzmann equation.

Population of energy level at thermal equilibrium condition:

At thermal equilibrium, the number of atoms (Population) at each energy level decreases exponentially with increasing energy level, as shown in the figure below.

Let us consider, two energy levels $E_{1}$ and $E_{2}$. The population of these energy levels can be calculated by

$N_{1}=e^{\left(-\frac{E_{1}}{kT} \right)} \quad (1)$

$N_{2}=e^{\left(-\frac{E_{2}}{kT} \right)} \quad (2)$

The ratio of the population in these two levels is called the relative population.

$\frac{N_{2}}{N_{1}}= \frac{e^{\left(-\frac{E_{2}}{kT} \right)}}{e^{\left(-\frac{E_{1}}{kT} \right)}}$

$\frac{N_{2}}{N_{1}}= e^{\left(-\frac{(E_{2}-E_{1})}{kT} \right)} $

$\frac{N_{2}}{N_{1}}= e^{-\frac{\Delta E}{kT} } $

This equation is known as Boltzmann's distribution. The above equation suggests the relative population is dependent on two factors.

1.) The energy difference $(\Delta E)$

2.) The absolute temperature $T$

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Ferromagnetic Substances and Its Properties

Description:

The atoms of these materials like paramagnetic material have permanent magnetic dipole moments. The similar natures of dipoles are grouped in a small region called domain. These domains have a net magnetic moment in a particular direction. In the material, there are a large number of domains having magnetic moments in different directions making the net magnetic moment of the entire material zero. When the external magnetic field is applied to such ferromagnetic materials, then either the domains are oriented in such a way as to align with the direction of the field, or the size of the favorable domain increases. Generally, in the strong applied field the domains are aligned and in the weak field the size of the favorable domain increases. In both cases, the material is strongly magnetized in the direction of the applied external magnetic field.

Properties of Ferromagnetic Substances:

The properties of ferromagnetic substances are similar to the properties of diamagnetic substances but the difference is that diamagnetic substances are weakly magneties and ferromagnetic substances strongly magenties in the presence of magnetic field.

Properties of Ferromagnetism:

1.) If these materials are placed in an external magnetic field, they are strongly magnetized in the direction of the applied external magnetic field.

2.) Due to unpaired electrons, the atoms of ferromagnetic materials have a net magnetic dipole moment.

3.) Ferromagnetism arises due to the formation of domains.

4.) Magnetic susceptibility of these materials is high and positive and also inversely proportional to the absolute temperature:

$\chi=\frac{C}{T-T_{c}}$

where $T_{c}$ is Curie temperature.

5.) Relative permeability of these materials is much greater than 1.

6.) Magnetic moment is high but along the direction of the applied magnetic field.

7.) $Fe$, $Ni$, $Co$, etc. are examples of paramagnetic materials.

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High Monochromaticity of Laser Light

High Monochromaticity:

A laser beam is highly monochromatic. The monochromaticity of the laser beam is much more than that of any traditional monochromatic source. The line spread of a laser beam is very small in comparison to the light from a traditional source. This difference arises because conventional sources emit wave trains of very short duration and length, whereas, laser emit continuous waves of very long duration. The random spontaneous emission in the laser cavity is one of the mechanisms that determine a laser's ultimate spectral line width. It should be noted that no light source including laser light source, is perfectly monochromatic but a better approximation to the ideal condition may be considered in the case of the laser beam. The spread of light from a normal monochromatic source range over a wavelength of the order of $100 -1000 \overset{\circ}{A}$ while in lasers it is of the order of few angstroms $(\lt 10 \overset{\circ}{A})$ only.

The high spectral purity of laser radıation leads directly to applications in basic scientific research including photochemistry, luminescence excitation spectroscopy absorption, Raman spectroscopy, and also in communication. The degree of non-monochromaticity $\xi$ of light is characterized by the spread in frequency of a line by the line width $\Delta \nu $ and is expressed as:

$\xi=\frac{\Delta \nu}{ \nu_{\circ}}$

where $\nu_{\circ}$ is the central frequency. If $\Delta \nu $ approaches zero the degree of non-monochromaticity tends to zero which is an ideal condition. Absolute monochromaticity $(\Delta \nu =0)$ is not attainable in practice even with laser light. The spreads of two light sources, laser light, and normal light, are shown in the figure above. The degree of non-monochromaticity may also be written in terms of coherence time $(\tau_{C})$ or coherence length $(L_{C})$ as follows:

$\xi=\frac{1}{\tau_{C} \: \nu_{\circ}}$

$\xi=\frac{c}{L_{C} \: \nu_{\circ}}$

This relation shows that the monochromaticity will be large for higher values of coherence time or coherence length. The bandwidth of a laser light from a high-quality He-Ne gas laser is of the order of $500Hz$ $(\Delta \nu =500 Hz)$ corresponding to coherence length of the order of $600 km$ $(\tau_{C} = 2 \times l0^{-3} sec)$.

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Conversion of Galvanometer into a Voltmeter

What is a voltmeter?

A voltmeter is an instrument used to measure potential differences between two points in an electric circuit directly in volts. The instrument measuring the potential difference of the order of millivolt $(mV)$ is called a millivoltmeter. An ideal voltmeter has infinite resistance.

Galvanometer used as voltmeter:

To use the galvanometer as a voltmeter in the circuit, The resistance of the galvanometer should be very high or almost infinite as compared to the other resistance of the circuit. Because the internal resistance of an ideal voltmeter is infinity.

So a high resistance is connected in series with the galvanometer (pivoted-type moving-coil galvanometer).

When a high resistance is connected in series to the galvanometer then the resultant resistance increases as compared to the other resistance of the circuit and it can be easily used as an ammeter and the actual potential difference can be measured through it.

Mathematical Analysis:

Let us consider, $G$ is the resistance of the coil of the Galvanometer, and the $i_{g}$ current, passing through it, produces full-scale deflection. If $V$ is the maximum potential difference that exists between two points $a$ and $b$ in the circuit. On connecting the galvanometer across the points $a$ and $b$, a current $i_{g}$ passes through the galvanometer and a high resistance $R$ is connected in series with galvanometer then

$i_{g}= \frac{V}{G + R}$

$G + R= \frac{V}{i_{g}}$

$R= \left(\frac{V}{i_{g}}\right ) - G$

If the current $i_{g}$ in the coil produces a full-scale deflection, then for the potential difference $V$ between the points $a$ and $b$, there will be a full scale deflection. Thus, on connecting a resistance $R$ of the above valve in series with the galvanometer, the galvanometer will become a voltmeter of range $0$ to $V$ Volt.

Note:

For the voltmeter, a high resistance is connected in series with the galvanometer and so the resistance of a voltmeter is very high compared to that of a galvanometer.

Resistance of voltmeter

$R_{v}=R+G$

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