Electric Charges and Fields Revision PDF

# Chapter-1 Electric Charges and Fields: Revision ## Electric Charge * **Charge is a property of matter which gives rise to electrostatic force** * **Symbol:** $q$ or $Q$ * **SI unit:** Coulomb ($C$) * **Dimension:** $[AT]$ * **Types of charges:** Positive (+) and Negative (-) ## Properties of Charge 1. **Charge is additive:** The total electric charge on an object is equal to the algebraic sum of all the electric charges distributed on the different parts of the object. $q = q_1 + q_2 + q_3 +.... + q_n$ 2. **Charge is quantized:** Charges are integral multiples of elementary charge 'e' $(= 1.6 \times 10^{-19}$ C). $Q = ne$ (where $n =0, \pm1, \pm2, \pm3...$) 3. **Charge is conserved:** For an isolated system, the net charge always remains constant. ## Coulomb's Law: * **Force between two charges is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them.** > $F \propto \frac{q_1 q_2}{r^2}$ * **K = 9 x 10^9 Nm^2/C^2** (This is the proportionality constant) > $F_{air} = \frac{Kq_1 q_2}{r^2}$ * **k = 1/4π εo** (This is also the proportionality constant) * **εo = 8.85 x 10-12 C2/Nm2** * **εo** → permittivity of air/vacuum ## Electric Field: * **It is the force experienced by a unit positive charge at a point.** * **Electric Field:** $\vec{E} = \frac{\vec{F}}{q}$ * **It is a vector quantity** * **SI Unit:** N/C ### Electric Field due to a Point Charge: * **E = kq/r2** (Point charge of q at a distance r from the point where we need to find the field) ## Electric Field Lines: * **Electric field lines are imaginary lines which represent Electric Field** * **Properties of E field lines:** * Field lines start from positive charges and end at negative charges. If there is a single charge they may start or end at infinity. * Two field lines can never intersect each other. If they did, the E Field at the point of intersection will have two directions, which is not possible. * The tangent to a line at any point gives the direction of E at that point. * The electric field lines can never form closed loops as a line can never start and end on the same charge. * In charge-free regions, electric field lines can be taken to be continuous curves without any breaks. * Electric Field lines are closer where E field is stronger and farther apart where E field is lesser * In a uniform field, the field lines are straight, parallel, and uniformly spaced. ## Dipole Moment: * **Electric Dipole:** A pair of equal and opposite charges separated by a small distance. * **Dipole Moment:** Product of magnitude of 1 charge and distance between charges. * **Direction:** Negative to positive * **SI Unit:** Cm * **P = distance x charge** * **P = 2aq** ### Field Due to Dipole: Axial Point: * **E+q = Kq/(r-a)2 (towards right)** * **E-q = Kq/(r+a)2 (towards left)** * **Enet = E+q + E-q** = Kq [1/(r-a)2 - 1/(r+a)2] $ Enet = Kq [ (r+a)2 - (r-a)2 ] / [ (r+a)2 (r-a)2 ] $ * **Enet = Kq [ 4ar ] / [ r^4 - a^2 ]^2** * **Assume r >>>>>> a, a2 ≈ 0** * **Enet = Kq [ 4ar ] / [ r4 ] = Kq(4a)/(r3) = (2aq) 2k/(r3) = 2kp/r3** * **Direction of Enet: Same as that of P** * **Eaxial = 2kp/r3** ### Field Due to Dipole: Equatorial Point: * **|E+q| = Kq/x^2** * **|E-q| = Kq/x^2** * **|E+q| = |E-q| = E = Kq/x^2** * **At P, vertical component of E field cancels** * **Enet = 2Ecosθ** * **Enet = 2 * kq/x2 * a/x = kp/x3** * **Enet = kp/(x2 + a2)3/2** * **Assume r >>>>>> a, a2 ≈ 0** * **Eeq = kp/r3** * **Direction will be opposite to P** * **Eeq = -kp/r3** * **Eaxial / Eeq = 2 / 1** ## Torque on Dipole in Uniform Electric Field: * **Fnet = 0 [Uniform Field] (The forces on the +q and -q charges are equal and opposite)** * **τ = F x 1r distance (τ is the torque)** * **τ = qE x 2a sinθ** * **τ = pE sinθ** * **τ = p x E** * **τ = 0 (min), θ=0° (Stable Equilibrium). At this angle, the electric field and dipole moment are parallel to each other.** * **τ = 0 (min), θ=180° (Unstable Equilibrium). At this angle, the electric field and dipole moment are antiparallel.** * **τ = pE (max), θ=90° (Maximum Torque). At this angle, the electric field and dipole moment are perpendicular to each other.** ## Electric Flux (Φ): * **Φ = E.S = E S cosθ** * **Φ = ∫E.dS** * **SI Unit: Nm2/C** ## Gauss's Law: * **Statement:** Gauss's Law states that the total flux through a closed surface is 1/εo times the net charge inside it. * **Mathematical Form:** ∫E→.dS→ = q/εo * **Φ = ∫E→.dS→** * **Φ = q/εo** ## Electric Field Due to: 1. **Infinite straight wire:** * **Sds = CSA = 2πrl** * **λ = q/l** * **E = λ / 2πεor** * **E α 1/r (Electric Field decreases as the distance from the wire increases)** 2. **Infinite charged sheet:** * **σ = q/A (σ is the surface charge density)** * **E = σ / 2εo** * **The Electric field is uniform (The electric field does not change with the distance from the sheet).** 3. **Spherical Shell/Hollow Sphere:** * **Inside shell: ∫E.dS = q/εo = 0 (q = 0)** * **Einside = 0** * **Outside shell: ∫E.dS = q/εo (θ = 0)** * **E ∫dS = q/εo** * **∫dS = 4πr2** * **E 4πr2 = q/εo** * **E = q / 4πεor2** * **E = Kq/r2 (Same as E due to a point charge)** * **E α 1/r2 (The Electric field decreases with the square of the distance from the sphere)** * **σ = q/A = q / 4πR2** * **q = σ 4πR2** * **The Electric field inside the sphere is zero.** ## Example Problems * **How many electrons constitute 1 coulomb of charge (e = 1.6 x 10-19 C)?** * **q = ne** * **1C = (n) e** * **n = 1C/e = 1C / 1.6 x 10-19C** * **n = 1019 x 0.625** * **= 6.25 x 1018 electrons** * **A polythene piece rubbed with wool is found to have a negative charge of 3 x 10-7 C. (a) Estimate the number of electrons transferred (from which to which)? (b) Is there a transfer of mass from wool to polythene? ** * **(a) q = ne** * **3 x 10-7 = ne)** * **n = (3 x 10-7 / e) = 3 x 10-7 / 1.6 x 10-19 = 3 x 10-7 x 1019 / 1.6 = 3 x 1012 / 1.6 = 1.875 x 1012 electrons** * **Electrons are transferred from wool to polythene** * **(b) e has mass me = 9.11 x 10-31 kg** * **total mass of e's transferred = (1.875 x 1012 ) (9.11 x 10-31)** * **= 17 x 10-19 kg** * **Can a body have a charge of 8 x 10-18 C?** * **q = ne** * **8 x 10-18 = (n) 1.6 x 10-19** * **n = 8 x 10-18 / 1.6 x 10-19 = 5 x 10 = 50** * **Since n is an integer, it is possible** * **What is the force between two small charged spheres having charges of 2 x 10-7 C and 3 x 10-7 C placed 30 cm apart in air?** * **F = kq1q2/r2** * **= 9 x 109 (2 x 10-7) (3 x 10-7) / (0.3 x 0.3)** * **= 54 x 10-5 / 0.09** * **= 600 x 10-5** * **= 6.00 x 10-3N** * **The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge -0.8 µC in air is 0.2 N. (a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first?** * **(a) F = kq1q2 / r2** * **r2 = kq1q2 / F = 9 x 109 (0.4 x 10-6) (0.8 x 10-6) / 0.2N** * **r2 = 14.4 x 10-3** * **r2 = 144 x 10-4** * **r = √(144 x 10-4) = 12 x 10-2 m = 12 cm** * **(b) The force on the second sphere due to the first is equal in magnitude but opposite in direction to the force on the first sphere due to the second sphere. Therefore the force on the second sphere due to the first is 0.2N. ** * **If force between two charges at 1 m separation is F, what is the force if the separation is 2m?** * **Technique:** If the distance is increased n-times, the force becomes 1/n2 times. * **Since the distance is increased by 2 times, the force will become (1/2)2** * **F → (1/2)2 = F/4** * **If force between two charges at 2m separation is F, what is the force if the separation is 6m?** * **The distance is increased 3 times (6m/2m = 3)** * **The force becomes 1/32 = 1/9 times** * **Therefore, Fnew = Fold / 9** ## Dielectric Constant: * **K = Eair / Eo = Em / Eo** * **Em = K . Eo** * **K = 1 for air** * **K > 1 for any other medium** * **8.85 x 10-12 = εo (permittivity of air)** * **Em** → permittivity of medium * **Fair = Kair q1q2/r2** * **Fair = 1/(4πεo) q1q2/r2** * **Fm = 1/(4πεoεm) q1q2/r2 = Fair/K** ## Charged Bodies: * **Neutral body:** Equal number of positive and negative charges. * **-vely Charged body:** More negative charges than positive charges. * **+vely Charged body:** More positive charges than negative charges. ## Charging Methods: ### Charging by Friction: * **Two uncharged insulators can acquire unlike charges on each on rubbing against one another. This is called charging by friction.** ### Charging by Contact: * **Charging a body by actual contact is called charging by contact / conduction.** ### Charging by Induction: * **Charging by induction does not require direct contact between the charged object and the object that is being charged.** ## Conductors and Insulators: * **Conductors:** There are large numbers of free electrons. * **Examples of Conductors:** Metals, Human Body, Earth * **Insulators:** Electrons are tightly bound to their atoms * **Examples of Insulators:** Glass, Plastic, Wood * **Semiconductors:** In between conductors and insulators. They have a limited number of free electrons. ## Gold Leaf Electroscope: * **It is used to detect the presence of charged bodies.** ## Earthing: * **Sharing of excess charge of a body with the earth is called earthing / grounding.** ## Important Notes * The **Coulomb's Law** equation describes the force between two point charges. The force is attractive if the charges have opposite signs and repulsive if they have the same sign. * **Electric field lines** map the direction of the electric field. They always start on positive charges and end on negative charges and never cross each other. * **Electric flux** is a measure of the amount of electric field passing through a surface. * **Gauss's Law** is a fundamental law of electrostatics that relates the electric flux through a closed surface to the net charge enclosed by the surface. It is used to calculate the electric field of charge distributions. * **Dielectric Constant** is a measure of the ability of a material to reduce the electric field strength. * The **torque** on a dipole due to a uniform electric field is proportional to the dipole moment and the sine of the angle between the dipole moment and the electric field. * **Conducting materials** allow current to flow easily because they contain free electrons. **Insulators,** however, prevent the flow of charge. **Semiconductors** are materials that have properties between conductors and insulators, and their conductivity can be manipulated. * The **gold leaf electroscope** is a device used to detect the presence of electric charge. * **Earthing** is the process of connecting a body to Earth, allowing the excess charge to flow to the ground. * In general, **charged bodies** interact with each other via electrostatic forces, which can be either attractive or repulsive. The force between two charges is stronger when they are closer together and weaker when they are farther apart.

Electric Charges and Fields Revision PDF

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