Physics for Class 12 Boards: Backbencher Series

Introduction to Backbencher Series

• The series aims to comprehensively cover 14 chapters with lectures delivered quickly.
• It is designed for students with backlogs.

Special Features of the Series

• Brief explanations
• Numerical examples
• Conceptual questions
• Derivations
• Theory

Additional Resources for Practice

• "Physics One Shot Question Bank"
• "Physics Question Bank."
• They are tailored for Class 12th board exams.

One Shot Book

• Each chapter is summarized in two pages using mind maps.
• Offers practice questions with complete solutions.
• Includes previous year questions (PYQs) along with important questions from NCERT and new conceptual questions.

Physics Question Bank

• Includes theory along with questions

How to Order

• Links for ordering are provided in the video description and comment section.

Winners Batch

• Offers free access to two batches for daily physics learning
• One batch covers syllabus ahead and the other behind, along with free marathon classes.

What is Charge?

• It's a property of a substance that causes it to exert electrostatic forces.
• Electrostatic force comes in two types: attraction and repulsion.
• Attraction occurs between opposite charges or between a charged and a neutral body.
• Repulsion always occurs between similar charged bodies.

Cause of Charge

• Charge arises when there is an imbalance in the number of electrons and protons within a substance or atom.
• A negative charge occurs when there is an excess of electrons.
• Mass increases.
• A positive charge occurs when there is a deficit of electrons.
• Mass decreases.

Properties of Electric Charge

• Invariance: Charge remains unaffected by changes in speed.
• Additivity: Charge follows algebraic addition, meaning direction doesn't affect it.
• Conservation: Charge cannot be created or destroyed, only transferred.
• Quantization: Charge is transferred in integral multiples from one body to another.
• Charge is always transferred in the form of packets.

Charge Distribution on Conductors

• When a conductor is charged, the charge resides on its surface not inside.
• Charge density is higher at sharp points.
• Charge density is inversely proportional to the radius of curvature.

Repulsion and Attraction Examples

• If body 'x' is positively charged and repelled by 'y', then 'y' must also be positively charged.
• If object 'z' is attracted to 'y', then 'z' could be negatively charged or neutral.

Coulomb's Law

• Force between two charges
• The formula for the force (F) between two charges (q1, q2) separated by distance (r) in vacuum is: F = k * q1 * q2 / r^2.
• 'k' is Coulomb's constant, valued at approximately 9 x 10^9 Nm²/C².
• Always use the magnitude of the charges in calculations.

Vector Form of Coulomb's Law

• Used to show force direction.

Numerical Problems based on Coulomb's Law

• Covers how to calculate net force on a charge.

Steps

• Calculate individual forces and their directions.
• Find the resultant force using vector addition.
• Important when charges are placed at the vertices of geometrical shapes.
• Example is calculating force on a charge at the vertex of an equilateral triangle due to other charges.
• Determining net force at a corner of a square due to other charges.

Limitations of Coulomb’s Law

• Only valid for static charges.
• Only valid for point charges.
• Only valid for distances larger than 10^-15 meters.

Important Graphs Related to Coulomb's Law

• In inverse square relationship, graph forms as a curve.
• In direct relationship, graph appears as straight line.
• Relationship between force and square of distance is inverse.
• Graphs of force versus 1/r^2 are always straight lines.

Key Points for Analyzing the Graph

• When q1*q2 is positive, the graph shows a repulsion force and is plotted above the axis.
• Greater product means steeper slope.
• When q1*q2 is negative, the graph shows an attraction force and the plot falls below the axis.
• Smaller product correlates to lesser slope.

Problem Solving in Electrostatics

• Focus is calculating net forces on point charges on a coordinate plane or geometrical shape
• Uses concepts like force addition, vector components etc.

Sharing of Electric Charge

• When two identical conducting spheres come into contact, charge will distribute evenly.
• If a charged sphere touches an uncharged sphere, charge is equally distributed.
• Process may be iterative, touching multiple spheres sequentially can lead to different charge distributions.

Electric Field

• The area around a charge where electrostatic force is experienced.
• Electric field intensity is the strength of the electric field.
• Formula: E = F/q0, where F is force and q0 is the test charge.
• The electric field is independent of the test charge.

Properties of Electric Field Lines

• Imaginary lines that represent the electric field.
• Tangent at any point indicates the direction of force on a positive charge.
• They never intersect because electric field only has one direction at any given point
• Always originate from positive charges and end on negative charges.
• Smooth and continuous curves that extend to infinity, until encountering a negative charge.
• Always perpendicular to the surface of a conductor to prevent surface current.
• Charge density is high

Patterns of Electric Field Lines

• For a positive charge, lines radiate outward.
• For a negative charge, lines point inward.
• For electric dipoles, lines extend from the positive to negative charge.
• Field lines around finite-length charged wire curve due to charge repulsion close to the ends.
• Infinite shape charged wires are in a straight line.
• When two equal charges present, they repel and form a neutral point between them
• Neutral point is closer to the smaller charge.

Special Case: Positive Charge and Neutral Conducting Plate

• The positive charge causes negative charge to accumulate on the side of the plate nearest the charge, and positive to repel to the other side.
• Electric field lines emanate from the positive charge and terminate on the negative charges induced on the plate.

Electric Dipole

• Pair of equal and opposite charges separated by a distance (2l).
• Dipole moment (p) is calculated as q * 2l.
• It's a vector that points from the negative to the positive charge.

Electric Field Due to a Dipole

• Axial Point: A point on the line of the dipole.
• Electric field formula on the axial point due to charge
• Equatorial Point: A point perpendicular to the line of the dipole.
• Electric field on equatorial point due to charge

Steps

• Electric field at axial point
• Find electric field due to positive charge, (k * q) / (r - l)^2.
• Find electric field due to negative charge, (k * q) / (r + l)^2.
• Subtract smaller absolute electrics field from larger one.
• Assuming r >> l, yields the final electric field as k * 2 * p / r^3.
• Electric field at equatorial point
• Electric field due to positive and negative charges are equal and opposite, resulting in cancellation of vertical components.
• Only the horizontal components get added, the effective electric field is k p / r^3.

Comparing Axial and Equatorial Fields

• Electric field is twice as strong at an axial point near the dipole when compared to equatorial point.
• Axial point is more strong than Equatorial point.
• Electric field decreases with distance at rate of 1/r^3 for dipoles, and 1/r^2 for point charges.

Electric Flux

• Number of electric field lines passing perpendicular through surface.
• Formula: Φ = E.A = e x A cos(θ), with cos(θ) is the angle between the electric field vector and area vector.
• When electric field is parallel to plane, electric flux is zero.
• When electric field is perpendicular to plane, electric flux is maximum.
• For enclosed shapes, the area vector points outwards.

Golden Point - Angle Clarification

• If angle given relative to the plane, subtract given angle from 90, theta=90 – Given angle
• If angle given normal to plane, use given angle directly.

Example Problems

• Calculating electric flux when given electric field strength and area.
• Determining flux and electric field at specified angles relative to plane area.

General Steps for Solving Complex Field and Flux Problems

• Find component of electric field
• Evaluate value component of area and vectors
• Establish angle relative to plane.

Gauss's Law

• Correlates electric flux through surface with enclosed charge.
• Formula: Φ = q (enclosed) / ϵ0.
• Only enclosed charge contributes to flux calculation.
• Shape charge doesn't matter

Gaussian Surface

• Surface around source to calculate electric flux.
• If point charge, use surface shape of sphere.
• If linear charge, use surface shape of cylinder.
• Surface designed to enclose system creating electric field.

Proof of Gauss Law

• At any point the net number of field lines penetrating an area is proportional to enclosed charge
• Consider field to be enclosed charge.
• Divide spherical area under consideration into infinitesmal surfaces “ds”.
• Take integral. Integrate over surface get closed equation by applying electric field.
• Electric field outside a uniformly charged thin shell will behave like point charge.

Applications of Gauss's Law

• Calculating electric field.
• Relationship between electric field and electric force.

Questions & Solving Strategies

• When finding flux through shapes inside each other start from inner shapes and work around the outside to consider net charge.
• If a dielectric inserted into space between shapes, consider effect on electric field based on inserting value.

Calculation Strategy

• Use formulas, integrate and vector properties.

Specific Geometry (Cube) for Charge Placement and Flux Encounter

• Focus now turned to cube based structures with 4 key types/locations of questions 1- center 2- vertex 3- side 4- faces/surfaces

Formulas By Charge Placement

• Center of cube -> Ф = q/epsilon naught
• Each of faces sees Ф q/(6 epsilon naught) amount of energy
• vertex of cube -> Ф = q/(8 epsilon naught)
• Only relevant cube sides 3 which are perpendicular to relevant
  • Ф = q/(24 epsilon naught)
• Side of cube Ф = q/(4 epsilon naught)
• Only relevant sides 2 which are perpendicular to relevant
• Ф = q/(24 epsilon naught)
• Surface -> Ф = q/(6 epsilon naught)

Key Data Note For Specific Vertex/Corner or Edge of Cubes.

• Can relate charge, placement, electric field vector
• Flux for given position
• Position near parallel vertex or corners.

Hemispheres

• Used as calculation tool by covering structure consider overall structure charge field
• If hemisphere at base or in middle then Ф = q/2 epsilon naught
• Due to 2x hemisphere covering.

Geometry

• Remember all vector formula from triangle and cube applications.

Thin Sheet

• Note if sheet is considered with side, at point charge or inside a side. Note vector to point charge and area of sides and apply integration.

DSB-SC Modulation

• DSB-SC is equivalent to वेट.ds0

Angle Considerations for Surface Integrals

• For surface s1, the angle θ is 0 degrees.
• For surfaces s2 and s3, the angle is 90 degrees.
• The values for s2 and s3 become zero due to the 90-degree angle.

Electric Field Calculation for a Cylinder

• The height of the cylinder is denoted as 'h' or 'l'.
• ∮ E.ds = q / ε0 simplifies to E (2πrl) = q / ε0 for a cylindrical surface.
• The final expression for electric field E = q / (2π ε0 rl).

Linear Charge Density

• Linear charge density (λ) is introduced when charge is distributed along a length.
• λ = q / l, where 'q' is the charge and 'l' is the length over which it is distributed.
• q = λl can be substituted into equations involving point charges for continuous charge distributions.
• Substituting q = λl, L cancels, leading to a modified electric field formula.

Electric Field and Distance Graph

• Electric field and distance have an inverse proportional relationship
• The graph of electric field vs charge density shows a proportional relationship.

Electric Field Due to a Charged Sheet

• A charged sheet has charge distributed across its surface.
• A cylinder is used to enclose a portion of the charge for electric field calculation.
• For the flat surfaces (s1, s2) of the cylinder, the angle θ = 0 degrees.
• For the curved surface (s3) of the cylinder, the angle θ = 90 degrees.

Applying Gauss's Law to the Charged Sheet

• Gauss's Law integral simplifies to ∫ E.ds = q / ε0
• The electric field on one side of a charged sheet is σ / 2ε0, where σ is the surface charge density.

Electric Field Between Two Charged Sheets

• If two positively charged sheets have the same charge density (σ)
• The electric field in between the sheets is zero.
• The electric field outside the sheets is σ / ε0 on both the right-hand side and left-hand side.

Electric Field Between Oppositely Charged Sheets

• If two sheets have opposite charges (+σ and -σ)
• The electric field between them is σ / ε0.
• The electric field is zero on the outside.
• This is applicable in capacitors.

Calculating Electric Field Due to a Charged Sphere

• Electric field is calculated at different locations around the sphere (outside, on the surface, inside).
• Gaussian surface is created.
• Due to symmetry, θ is always 0 degrees.
• Radius of sphere is Capital R, Gaussian surface is small r

Electric Field Outside the Charged Conducting Sphere

• Gaussian surface encloses the sphere (r > R).
• Gauss’s Law: ∮ E. ds = q / ε0.
• Solving ∮ E.ds for a sphere yields E (4π r^2) = q / ε0.
• Electric field outside conducting sphere is E = q / (4π ε0 r^2).

Electric Field on the Surface of the Charged Conducting Sphere

• Gaussian surface coincides with the sphere's surface (r = R).
• Replace "r" from the previous formula with sphere Radius "R".
• Substituting q = σA = σ (4π R^2) in Gauss's Law.
• After simplifying, electric field at surface of a conducting sphere is E = σ / ε0 (maximum electric field).

Electric Field Inside the Charged Conducting Sphere

• Gaussian surface inside the sphere (r < R) encloses no charge.
• Net charge inside the Gaussian surface = 0
• Electric field inside is zero

Electric Field Plot

• Electric field inside conductor is zero
• Electric field on the surface is maximum
• Shows E vs. r for conducting shell or sphere.
• The field increases to a maximum at r = R.
• Beyond the sphere, the electric field decreases with distance as 1/r².
• Plot the relationship between charge and distance

Torque on a Dipole in an Electric Field

• A dipole in an electric field experiences a torque
• Electric field exerts a force on positive charge (F = qE) & negative charge
• The forces cause the dipole to rotate.

Torque Calculation

• Torque = Magnitude of force × Perpendicular distance between forces.
• Perpendicular distance between forces is calculated usign trigonometry, it turns out to be 2l sin(θ)
• Torque is expressed as τ = pE sin(θ).
• In vector form, torque is given by τ = p × E.

Special Cases

• Torque is maximum (pE) when θ = 90 degrees.
• Torque is minimum (zero) when θ = 0 or 180 degrees.
• θ is the angle between the dipole moment (p) and electric field (E).
• Dipole moment direction is from negative to positive charge.

Equilibrium Types

• Dipoles have stable equilibrium at 0 degrees.
• Dipoles have unstable equilibrium at 180 degrees.

Non-Uniform Electric Field Effects

• Torque and linear force is experienced
• Non-uniform field causes rotational and linear motion.

Equilibrium of Three Charges

• Equilibrium point is at midpoint
• Two same sign and a point charge, the third charge is placed inbetween (assuming attractive forces in different directions)
• Two opposite sign point charges, the third charge is placed closes to the smaller magnitude charge
• The position is located depending on force equations and net force

Steps to solve 3 point charge equilibrium

• Given are charges q and 2q, separated by a distance r.
• Determine where a third charge q0 should be placed for the system to be in equilibrium.
• Equilibrium is achieved when q0 is between the charges(same nature charges), close to smaller charges.
• Coulomb's Law equations are used/calculated and used for comparison.
• Solve the equation for ‘x’ to find the position of q0.

Comprehensive physics series covering 14 chapters with concise lectures, numerical examples, and conceptual questions. Includes derivations and theory. Complemented by 'Physics One Shot Question Bank' and 'Physics Question Bank' for Class 12th board exam preparation.