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Electrostatics

• Physics that deals with charges at rest is called Electrostatics and that deals with moving charges is called Electrodynamics.
• An electric force is the force which holds the positive and negative charges the make up atoms, molecules and bodies.

Charges & its Relationship with Friction

• Bodies get charge due to friction.
• During rubbing process, free or atomic electrons gain more energy and get detached thus producing positive charge.
• Similar charges repel each other and opposite charges attract each other.
• Excess of electrons create negative charge while deficiency produces positive charge in an object.
• SI unit of charge is coulomb. it is defined as:

Coulomb

If force between two equal and opposite charges placed in a vacuum 1 m apart is \(9\times10^9N\) then each of them bear 1C charge.

OR

Coulomb

charge of \(6.25\times10^{18}\) electrons is 1C.

• Charge is conserved quantity as well as quantized.
• Charge per unit area \((\Delta Q/ \Delta A)\) is called surface charge density.
• Charge per unit volume \((\Delta Q/\Delta V)\) is called Volume charge density
• Charge per unit length \((\Delta Q/\Delta L)\) of straight conductor is called linear charge density.
• When charged particle is subjected to electrical field, then its acceleration is \(qE/m\).
• When charge particle is given to hollow sphere, it resides on outer surface only but not on inner surface.
• We can charge a particle by two methods:
  1. Induction (without physical contact).
  2. Conduction (with physical contact).

Electrical Forces

Coulomb found that electrical force between two point charges is:

1. Directly proportional to product of magnitude of the charges.
2. Inversely proportional to square of distance between them.
3. Act along line joining them.

Mathematically, $$F_e=k\frac{q_1q_2}{r^2} $$ Where, \(k=9\times10^9 Nm^2 / C^2\)

In Vacuum, and using S.I units $$ k = 1 / 4\pi\epsilon_{\omicron} $$ Where, \(\epsilon_{\omicron}\) is permittivity of free space. $$ \epsilon_{\omicron} = 8.85 \times 10^{-12} C^2 / Nm^2 $$

• \(\overline{F_{12}} = -\overline{F_{21}}\) shows that electrical forces form an action-reaction pair.
• Coulomb's law obeys inverse square law.

• When charges are placed in medium other than air, then force reduces by a factor \(\epsilon_{r}\) known as relative permittivity of the medium. $$ F' = \left(\frac{1}{\epsilon_r}\right)F $$

• In terms of electrical force, \(\epsilon_r\) can be given as: $$ \epsilon_r = \left(\frac{F}{F'}\right)$$

• \(\epsilon_r\) is a ratio and hence unitless.

Info

Two charges are always taken as point charges because they are very small in size as compared to the distance between them.

Electric Field

Electric Field is a vector quantity.

• It is intrinsic property of charge to have electric field around it.
• Two theories has been put forward to explain electric field:
  1. Action at a distance (Newton's view)
  2. Field theory (Faraday's view)
• Action-at-distance has been rejected experimentally while field theory is a convincing view.
• For infinite extent, field is uniform all over.
• For finite extent, field is not uniform at ends. (Fringing Field)

Electric Intensity

• Electric Intensity is given as: $$ E = F / q_{\omicron} $$ Where \(q_{\omicron}\) is a test charge. it is defined as the force per unit positive charge.
• In vector form $$\overline{E} = k \frac{q}{r^2} \hat{r} $$ for point charge.
• SI unit of \(E\) is \(N/C\) and \(V/m\)

Electric Field Lines

The path followed by a tiny positive charge in an electrical field are called Electric Field Lines.

• Originate from positive charge.
• End on negative charge.
• Do not intersect
• Contract longitudinally.
• Repel transversely.
• No electrical line is present inside the conductor.
• Tangent drawn to electric lines gives the direction of electric intensity.
• Electric Field is stronger where the electric lines are closely packed

Applications of Electrostatics

Xerography (Photocopier)

• "Xeros" means "dry", "graphy" means "writing"
• The lamp transfers the image of document as bright and dark spots on drum
• Drum is an aluminum cylinder coated with selenium. Selenium is a LDR. It retains positive charge in dark.
• Positive charge document image is created on the drum.
• Toner is given negative charge and is sprayed over the drum.
• The image is then transferred on paper and is settles by the heated pressure roller.

Inkjet Printers

• It works on the thin stream of ink ejected from the nozzle at high speed.
• It passes through two electrical components. Charging electrode and deflecting plates.
• Charging electrode gives a net charge to the stream controlled by instructions from computer.
• The uncharged drops pass without deflection through deflection plates and strike the paper while charged drops are directed to gutter.

Electric Flux

Electric Flux is the number of electric lines of force passing perpendicular through certain area.

Mathematically, $$ \phi_e = \overline{E} \cdotp \overline{A} = EA \cos \theta $$ provided \(\overline{E}\) is uniform on the flat surface represented by \(\overline{A}\).

• Its Scalar quantity.
• Flux is maximum when \(\theta=0^{\circ}\) (surface is perpendicular)
• Flux is zero when \(\theta=90^{\circ}\) (surface is parallel)
• If no. of flux lines leaving a closed surface is greater than no. of flux lines entering then surface contains source or positive charge.
• If no. of flux lines leaving a closed surface is less than no. of flux lines entering then surface contains sink or negative charge.
• If no. of flux lines leaving a closed surface is equal to no. of flux lines entering then surface contains neither sink nor source.

Info

Electric flux depends upon charge and medium not on the shape of the closed surface

Flux Through Closed Surface

• Flux through closed surface is: $$ \phi_e = Q / \epsilon_{\omicron} $$ It shows that flux through closed surface is independent of location of charges enclosed by it and shape of closed surface.

Question

Any apparatus placed within a metal enclosure is shielded from electric field. Why?

Gauss's Law

The flux through a closed surface is \(\left(\frac{1}{\epsilon_{\omicron}}\right)\) times the total charge enclosed in it.

Mathematically $$ \phi_{total} = \left(\frac{1}{\epsilon_{\omicron}}\right)Q $$

Gauss's Law is applied to calculate electric intensity. for this purpose an imaginary closed surface called Gaussian Surface is considered which must pass through the point at which the electric intensity is to be evaluated.

Applications

• Electric intensity inside a hollow but uniformly charges sphere is zero \(E = 0\)
• Electric intensity due to infinite sheet of charge is given as: $$ \overline{E} = \frac{\sigma}{2\epsilon_{\omicron}}\hat{r} $$
• Electric intensity between two equal but oppositely charged plates is given by $$ \overline{E} = \frac{\sigma}{\epsilon_{\omicron}}\hat{r} $$

Electric Potential & Potential Difference

Work done per unit positive charge in moving it against electric field with uniform velocity i.e. keeping the charge in electrostatic equilibrium.

Mathematically $$ \Delta V = \frac{W_{A\rightarrow B}}{q_{\omicron}} $$

• It can be given in term of potential energy as $$ \Delta V = \frac{\Delta U}{q_{\omicron}} $$ where \(W_{A\rightarrow B}=\Delta U =\) electrostatic P.E.
• SI unit of potential difference is volt.

If 1J of work is done on one coulomb charge between two points keeping the equilibrium, the potential difference is 1V. $$ 1V = 1J / 1C $$

Info

We can relate electric potential difference and electric field intensity by following relation: \(E=-\frac{\Delta V}{\Delta r}\)

Where negative sign shows that E is along decreasing potential \(N/C\) is equal to \(V/m\).

It means that:

• We can call \(E\) as potential gradient because it represents the maximum rate of change of potential difference w.r.t displacement.
• SI unit of \(E\) is \(N/C\) is equivalent to \(V/m\).

• Absolute electric potential at a distance \(r\) from source is given as: $$ V = \frac{1}{4\pi\epsilon_{\omicron}}\cdotp\frac{q}{r} $$
• Absolute electric potential at a point due to collection of point charges is given by $$ V = \frac{1}{4\pi\epsilon_{\omicron}}\displaystyle\sum_{i=1}^n\frac{q_i}{r_i} $$
• ECG records the voltage between points on human skin generated by electric process in the heart while EEG records that by brain.

Equipotential Surfaces

The surface on which electric potential is same at each of its points.

Example Surface charged hollow sphere containing point charge at its center

Characteristics of Equipotential Surfaces

• They do not intersect
• Potential difference between two points on equipotential surface is zero
• No work is done to move a point charge on an equipotential surface.
• Work is done when point charge is moved from one equipotential surface to another.

Electron Volt

The amount of energy acquired or lost by an electron as it traverses a potential difference of one volt

• \(1eV=1.6\times10^{-19}J\)
• \(1J=6.25\times10^{18}eV\)
• It is the unit of energy specially used for atomic particles.

Comparison of Gravitational & Electrostatics Forces

Particulars	Gravitational Force	Electrostatics Force
Formula	\(F_g=Gm_1m_2/r^2\)	\(F_e=kq_1q_2/r^2\)
Range	Infinite	Infinite
Symbol of Constant	\(G\)	\(k=\frac{1}{4\pi\epsilon_{\omicron}}\)
Value of Constant	Very Small	Fairly large
Nature	Always attractive	Attractive or repulsive
Dependence	Medium independent	Medium dependent
Relative strength	Weak: can be felt with massive object	Strong at close range

Millikan's Method for Electron' Charge

• Atomizer sprays oil droplets
• The droplet between the two plates could be suspended in air if the gravitational force \(F_g=mg\) acting downward on the droplet is equal to the electrical fore \(F_e=qE\) acting upward. \(qE=mg\) $$ E = \frac{V}{d}\;\;\implies\;\;q=\frac{mgd}{V}$$
• The terminal velocity is determined by timing the fall of the droplet over a measure of distance. Minimum quantum charge of electron is \(1.6\times10^{-19}C\).

Capacitor & Capacitance

Capacitor is a device used for storing electric charge and electrical energy.

• Charge stored by capacitor is given by: $$ Q=CV $$ Where \(C\) is the capacitance of capacitor.
• Capacitance is defined as:

Ability of a capacitor to store charge.
OR
The ratio of charge stored to the potential difference between plate of capacitor. $$ C=\frac{Q}{V} $$

• SI unit of capacitance is Farad.
• 1 Farad is defined as:

The capacitance of capacitor is one farad if a charge of one coulomb, given to one of the plate of parallel plate capacitor, produces a potential difference of one volt between them. $$ 1F= \frac{1C}{1V} $$

• Capacitance of parallel plate capacitor with air between its plates as: $$ C_{vac}= \epsilon_{\omicron}\frac{A}{d} $$ The above expression shows that:
  1. As we increase the area of plate the capacitance will increase.
  2. Decreasing the distance between the plates will increase the capacitance.
  3. Introducing a dielectric between the plates will increase the capacitance of the capacitor.
  4. Capacitor of an isolated charge sphere of radius \(R\) is $$ C= 4\pi\epsilon_{\omicron} R $$.
  5. Capacitance of a parallel plate capacitor with dielectric between its plate is given by: $$ C_{med}= \epsilon_r \epsilon_{\omicron} \frac{A}{d}= \epsilon_rC_{vac}\;\; \implies \;\; C_{med}>C_{vac} $$

Types of capacitor

• Parallel Plate Capacitor
• Spherical Capacitor
• Miniature Capacitor
• Tubular Capacitor
• Paper Capacitor
• Electrolytic Capacitor
• Variable Capacitor

Dielectric Coefficient/Constant

The ratio of the capacitance of a parallel plate capacitor with an insulating substance as medium between the plates to its capacitance with vacuum as medium. $$ \epsilon_r=\frac{c_{med}}{C_{vac}} $$

• Charging of capacitor is due to electrostatic induction phenomenon.
• Old name of capacitor is condenser.

Combinations of Capacitors

Series Grouping	Parallel Grouping
Capacitor are said to be connected in series between two points when we can proceed from one point to the other only through one path	The capacitors are said to be connected in parallel between any two points if we proceed from one point to the other along different paths.
In series grouping charge on each capacitor remains same and equals to the main charge supplied by the battery but potential difference across them may or may not be same.	In parallel grouping potential difference while charge on them may or may not be same.
Charge on each capacitor remains same and equals to the main charge supplied by battery. $$ V= V_1+V_2+V_3 $$	Potential difference across each capacitor remains same and equals to the main potential difference supplied by battery. $$ Q= Q_1+Q_2+Q_3 $$
Equivalent capacitance $$ \frac{1}{C_{eq}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}} $$	Equivalent capacitance $$ C_{eq} = C_1 +C_2+C_3 $$
In series combination potential difference and energy distribution is in the reverse ratio of capacitance. \(V \propto \frac{1}{C}\) and \(\text{P.E}\propto\frac{1}{C}\)	In parallel combination charge and energy distribution in the ration of capacitance. \(Q\propto C\) and \(\text{P.E} \propto C\)
If n identical capacitors each having capacitance \(C\) connected in series with supplied voltage \(V\) then equivalent capacitance \(C_{eq}=\frac{C}{n}\)	If n identical capacitors are connected in parallel \(C_{eq}=nC\)

Electric Polarization

Polarization is a phenomenon of appearance of charges on surface of the material subjected to electric field.

• Polarization charges are called induced charges.
• Dipole is a set of two equal and opposite charges separated by a small distance \(r\).
• Some substances are polar i.e. they have permanent dipoles e.g. in \(\text{NaCl}\), the end with \(\text{Na}\) ion is positive while that with \(\text{Cl}\) ion is negative.
• Some substances are non-polar i.e. they do not have permanent dipoles e.g. plastic. They become Polarization temporarily when subjected to external field.
• Polar molecules experience torque when subjected to uniform electric field but zero the force. In non-uniform electric field it also experience net force in addition to torque.

Dielectric

• Dielectric are non-polar insulators.
• Polarized dielectrics produce electric field opposite to the applied field.
• Molecular view of a polarized dielectric is shown below:

Where \(k=\epsilon_r\)

It shows that a static equilibrium exists within the dielectric

Info

Value of \(\epsilon_r\) for all doelectrics is greater than 1 expect for air.

Effect of Dielectric on Various Electric Quantities

\(F'=F/\epsilon_r\) (Decreases)

\(E'=E/\epsilon_r\) (Decreases)

\(V'=V/\epsilon_r\) (Decreases)

\(C'=C\epsilon_r\) (Increases)

Effect of Polarization on Capacitor

• The electric Polarization of dielectric increases the capacitance.
• The electrons in the dielectric remains bounded to their respective atoms. They just displaced from their normal positions.
• The molecules of the dielectric under the action of electric field become dipoles and dielectric is polarized.
• As \(E=\sigma/\epsilon_{\omicron}\), so electric field between Capacitor plates decreases due to polarization of dielectric.
• Decreases in \(E\) decreases the potential difference because \(V=Ed\).
• As a result capacitance increases.

Energy Stored in a Capacitor

• Charge on the plate of capacitor possesses electrical potential energy because of the work done to deposit the charge on the plates.
• \(\text{P.E} = \frac{1}{2}qV\)
• \(\text{Energy in capacitor}=\) \(\frac{1}{2}CV^2=\frac{1}{2}\frac{q^2}{C}\)
• Energy is stored in electric field between the plates.
• \(\text{Energy} =\) \(\frac{1}{2}\left[ \frac{A\epsilon_r\epsilon_{\omicron}}{d}\right] (Ed)^2\)
• \(\text{Energy density} =\) \(\text{Energy}/\text{Volume}\) \(= \frac{1}{2}\epsilon_r\epsilon_{\omicron}E^2\)

Info

• For D.C capacitor behaves as open circuit.
• For A.C capacitor behaves as closed circuit.

Charging and Discharging a Capacitor

• D.C supply stores charges on the plate
• A.C supply does not store charge.
• Charging and discharging time depends upon product of \(R\) & \(C\).

Time Constant

$$ t=RC $$ its unit is \(s\) (second).

• Time constant is defined as the time required by a capacitor to charge up to 0.63 times equilibrium charge on the capacitor. $$ t=0.63q_{\omicron} $$
• charge reaches its equilibrium value sooner when time constant is smaller.
• Windshield wiper of car works by charging and discharging of capacitor.

Effects of Separation on Different Factor

Separation is Increasing

Quantity	Battery removed	Battery connected
Capacitance	Decreases $$ C=\frac{1}{d} $$	Decreases $$ C=\frac{1}{d} $$
Charge	Remain constant because a battery is not present	Decreases because battery is present
Potential difference	Increases $$ V=\frac{q}{C} $$	Remains same as battery maintains the potential difference
Electric field	Remains constant $$ E=\frac{q}{A\epsilon_{\omicron}} $$	Decreases $$ E=\frac{q}{A\epsilon_{\omicron}} \implies E\propto q $$
Energy	Increases $$ U=\frac{q^2}{2C} $$	Decreases $$ U=\frac{1}{2}CV^2 $$

Separation is Decreasing

Quantity	Battery removed	Battery connected
Capacitance	Increases $$ C=\frac{1}{d} $$	Increases $$ C=\frac{1}{d} $$
Charge	Remain constant because a battery is not present	Increases because battery is present
Potential difference	Decreases $$ V=\frac{q}{C} $$	Remains same as battery maintains the potential difference
Electric field	Remains constant $$ E=\frac{q}{A\epsilon_{\omicron}} $$	Increases $$ E=\frac{q}{A\epsilon_{\omicron}} \implies E\propto q $$
Energy	Decreases $$ U=\frac{q^2}{2C} $$	Increases $$ U=\frac{1}{2}CV^2 $$

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