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Questions and Answers
Express Coulomb's law in vector form for the force applied by $q_1$ on $q_2$, denoted as $\vec{F}_{12}$.
Express Coulomb's law in vector form for the force applied by $q_1$ on $q_2$, denoted as $\vec{F}_{12}$.
$\vec{F}_{12} = \frac{kq_1q_2}{|\vec{r}_1 - \vec{r}_2|^3} (\vec{r}_1 - \vec{r}_2)$
The forces $\vec{F}{12}$ and $\vec{F}{21}$ are equal and opposite.
The forces $\vec{F}{12}$ and $\vec{F}{21}$ are equal and opposite.
True (A)
Derive an expression for the electric field $E$ at any point on the axial line of an electric dipole, where $q$ is the charge, $l$ is half the length of the dipole, and $x$ is the distance from the center of the dipole to the point.
Derive an expression for the electric field $E$ at any point on the axial line of an electric dipole, where $q$ is the charge, $l$ is half the length of the dipole, and $x$ is the distance from the center of the dipole to the point.
$E = \frac{2kPx}{(x^2 - l^2)^2}$
What is the electric field $E$ for an ideal dipole ($l << r$)?
What is the electric field $E$ for an ideal dipole ($l << r$)?
Derive an expression for the electric field $E$ at a point on the equatorial line of an electric dipole.
Derive an expression for the electric field $E$ at a point on the equatorial line of an electric dipole.
For an ideal dipole ($l << r$), what is the electric field $E$ on the equatorial line?
For an ideal dipole ($l << r$), what is the electric field $E$ on the equatorial line?
Deduce the expression for torque acting on a dipole of dipole moment $P$ in the presence of a uniform electric field $E$.
Deduce the expression for torque acting on a dipole of dipole moment $P$ in the presence of a uniform electric field $E$.
What is the torque when $\theta = 0$ (stable equilibrium)?
What is the torque when $\theta = 0$ (stable equilibrium)?
What is the torque when $\theta = 180^\circ$ (unstable equilibrium)?
What is the torque when $\theta = 180^\circ$ (unstable equilibrium)?
What is the torque when $\theta = 90^\circ$?
What is the torque when $\theta = 90^\circ$?
State Gauss's law, relating the net electric flux through a closed surface to the enclosed charge.
State Gauss's law, relating the net electric flux through a closed surface to the enclosed charge.
Derive an expression for the electric field $E$ due to a straight, uniformly charged infinite line, where $\lambda$ is the linear charge density.
Derive an expression for the electric field $E$ due to a straight, uniformly charged infinite line, where $\lambda$ is the linear charge density.
Find the electric field $E$ due to a uniformly charged infinite large plane thin sheet with surface charge density $\sigma$.
Find the electric field $E$ due to a uniformly charged infinite large plane thin sheet with surface charge density $\sigma$.
Derive an expression for the electric potential $V$ due to a point charge $q$ at a distance $r$.
Derive an expression for the electric potential $V$ due to a point charge $q$ at a distance $r$.
Derive the expression for the electric potential due to an electric dipole at any point on the axial line.
Derive the expression for the electric potential due to an electric dipole at any point on the axial line.
What is the net potential $V_{net}$ when $x >>> l$?
What is the net potential $V_{net}$ when $x >>> l$?
Determine the Potential Energy of an electric dipole in a uniform electric field.
Determine the Potential Energy of an electric dipole in a uniform electric field.
Deduce the expression for energy stored in a capacitor and also find the energy density.
Deduce the expression for energy stored in a capacitor and also find the energy density.
What is the equation for Energy density?
What is the equation for Energy density?
Obtain the expression for drift velocity of electrons in a conductor.
Obtain the expression for drift velocity of electrons in a conductor.
Using Kirchoff's rule obtain a balance condition in a Wheatstone Bridge!
Using Kirchoff's rule obtain a balance condition in a Wheatstone Bridge!
What is the principle of a Meterbridge? How can the ranknown resistance of a conductor can be determined?
What is the principle of a Meterbridge? How can the ranknown resistance of a conductor can be determined?
Derive an Expression for a potentiometer which is used to determine Internal Resistance of a cell.
Derive an Expression for a potentiometer which is used to determine Internal Resistance of a cell.
Using Biot-Savart Law, derive the expression for the magnetic field due to a circular coil carrying current at a point along its axis.
Using Biot-Savart Law, derive the expression for the magnetic field due to a circular coil carrying current at a point along its axis.
A long Solenoid of length l having N turns & carries current I. Deduce the expression at the center of a straight Solenoid.
A long Solenoid of length l having N turns & carries current I. Deduce the expression at the center of a straight Solenoid.
Using Ampere's circuital law, obtain the Magnetic field inside a Toroid (outside / Between).
Using Ampere's circuital law, obtain the Magnetic field inside a Toroid (outside / Between).
Obtain the expression for the Helical path of a charged Particle in a magnetic field?
Obtain the expression for the Helical path of a charged Particle in a magnetic field?
Obtain the expression for the force between 2 Parallel current-carrying conductors?
Obtain the expression for the force between 2 Parallel current-carrying conductors?
Deduce the expression for magnetic dipole momen of an electron orbiting around the nucleous.
Deduce the expression for magnetic dipole momen of an electron orbiting around the nucleous.
Explain Motional EMF & Deduce it's expression by the Concept of lorentz force?
Explain Motional EMF & Deduce it's expression by the Concept of lorentz force?
Derive an expression for the Mutual Inductance of 2 long coaxial solenoid of same length wound over each other.
Derive an expression for the Mutual Inductance of 2 long coaxial solenoid of same length wound over each other.
Deduce the expression for an Inductor, when the voltage is ahead of current by $\pi/2$ in pahse?
Deduce the expression for an Inductor, when the voltage is ahead of current by $\pi/2$ in pahse?
Deduce the expression for a capacitor when the voltage lag current.
Deduce the expression for a capacitor when the voltage lag current.
Write the expression for Impedence offered by series of LCR connected in AC source.
Write the expression for Impedence offered by series of LCR connected in AC source.
Obtain the relation between the critical angle of incidence & refractive index of medium.
Obtain the relation between the critical angle of incidence & refractive index of medium.
Obtain the expression for Refraction at a single refracting surface!
Obtain the expression for Refraction at a single refracting surface!
Derive lenz makes formula?
Derive lenz makes formula?
Derive the relation for refractive in team of angle of Deviation & angle of prism?
Derive the relation for refractive in team of angle of Deviation & angle of prism?
Using Huygen's principle to show, Houe a plane wavefront propogates from a rares to denser medium? Hence, verify snell's law of refraction.
Using Huygen's principle to show, Houe a plane wavefront propogates from a rares to denser medium? Hence, verify snell's law of refraction.
Using Huygen's principle, verify laws of reflection.
Using Huygen's principle, verify laws of reflection.
In Young's double slit, describe how bright & dark fringes are. obtained on the screen kept in front of a double slit? Obtain the expression for fringe width?
In Young's double slit, describe how bright & dark fringes are. obtained on the screen kept in front of a double slit? Obtain the expression for fringe width?
Single Slit experiment / Fraunhofer Diffraction!
Single Slit experiment / Fraunhofer Diffraction!
Derive the Expression for de-Broglie Equation?
Derive the Expression for de-Broglie Equation?
Flashcards
Coulomb's Law (Vector Form)
Coulomb's Law (Vector Form)
Coulomb's Law in vector form describes the electrostatic force between two point charges, indicating both magnitude and direction.
F₁₂ (Coulomb's Law)
F₁₂ (Coulomb's Law)
Force exerted on charge 1 due to charge 2.
F₂₁ (Coulomb's Law)
F₂₁ (Coulomb's Law)
Force exerted on charge 2 due to charge 1. It's equal and opposite to F₁₂.
Electric Field (Axial Line)
Electric Field (Axial Line)
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Enet (Electric Dipole)
Enet (Electric Dipole)
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E₂ (Electric Dipole)
E₂ (Electric Dipole)
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E₁ (Electric Dipole)
E₁ (Electric Dipole)
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Dipole Moment (p)
Dipole Moment (p)
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Electric Field (Ideal Dipole)
Electric Field (Ideal Dipole)
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Vnet (Ideal Dipole)
Vnet (Ideal Dipole)
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τ (Torque on Dipole)
τ (Torque on Dipole)
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Wel (Work Done on Dipole)
Wel (Work Done on Dipole)
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U (Potential Energy of Dipole)
U (Potential Energy of Dipole)
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U (minimum)
U (minimum)
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U (θ=90°)
U (θ=90°)
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Energy Stored in Capacitor
Energy Stored in Capacitor
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dW (Capacitor)
dW (Capacitor)
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U (Capacitor)
U (Capacitor)
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Energy Density (u)
Energy Density (u)
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u, expression for energy density
u, expression for energy density
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Drift Velocity
Drift Velocity
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Average Thermal Velocity
Average Thermal Velocity
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Net Force on Dipole (Uniform Field)
Net Force on Dipole (Uniform Field)
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Potential Difference
Potential Difference
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Electric Force
Electric Force
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Electric Potential and Field
Electric Potential and Field
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Capacitance (C)
Capacitance (C)
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Electric Field
Electric Field
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Work Done (Electric Field)
Work Done (Electric Field)
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Electric Field vs. Distance
Electric Field vs. Distance
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Study Notes
Coulomb's Law in Vector Form
- The force (F₁₂) applied by charge q₁ on charge q₂ is: F₁₂ = kq₁q₂*(r₁-r₂)/|r₁-r₂|³
- The force (F₂₁) applied by charge q₂ on q₁ is: F₂₁ = kq₁q₂*(r₂-r₁)/|r₂-r₁|³
- Since r₂ - r₁ = -(r₁ - r₂), therefore F₂₁ = -F₁₂
Electric Field of an Electric Dipole on Axial Line
- The net electric field (Enet) at point P is: Enet = E₂ - E₁
- Electric field due to +q: E₂ = k*q/(x-l)²
- Electric field due to -q: E₁ = k*q/(x+l)²
- Enet simplifies to: Enet = k*q * 4xl / (x²-l²)²
- Since p = q*2l, E = 2kpx / (x²-l²)²
- For an ideal dipole where l << r, l² is negligible: E = 2kp / x³ or E = 1/(4πε₀) * 2p / x³
Electric Field on Equatorial Line of an Electric Dipole
- Electric field magnitudes are equal: E₁ = E₂ = E
- The net electric field: Enet = 2Ecosθ
- Substituting cosθ = l/r: Enet = 2 * k*q/(x²+l²) * l/√(x²+l²)
- Which simplifies to: Enet = kp / (x²+l²)^(3/2)
- For an ideal dipole where l << r, l² is negligible: E = kp /x³ or E = 1/(4πε₀) * p / x³
Torque on a Dipole in a Uniform Electric Field
- Torque (τ) which equals force (F) times perpendicular distance (d): τ = F x d
- Total force: F = F₁ + F₂
- Fxd + Fxd = qElsinθ + qElsinθ = 2qElsinθ
- Since dipole moment p = q * 2l, torque simplifies to: τ = pEsinθ
- In vector form: τ = p x E
- The angle between p and E is θ
- Case I: When θ = 0°, τ = pEsin(0°) = 0, which is stable equilibrium
- Case II: When θ = 180°, τ = pEsin(180°) = 0, which is unstable equilibrium
- Case III: When θ = 90°, τ = pEsin(90°) = pE, which is the maximum torque
Gauss's Law
- Net electric field through a closed 3D surface is 1/ε₀ times the net charge enclosed by the surface: ∮closed = qin / ε₀ = ∮ E · dA
- Electric flux: dΦ = E · dA = EdAcosθ
- In a spherical surface, E is the same everywhere, so: ∮dΦ = ∮E · dA
- This simplifies to Φ = EA
- Φ = (1 / 4πε₀) * q / R² * 4πR²
- This simplifies to Φ = q / ε₀
Electric Field due to a Uniformly Charged Infinite Line
- λ is linear charge density, defined as λ = q/l
- The Gaussian surface is cylindrical
- Electric field is radially outward
- Net flux: Φnet = Φ₁ + Φ₂ + Φ₃
- Electric flux is calculated as: Φ = EA
- Total curved area of the surface is 2πrl and thus: Φ = El = E * 2πrl
- Electric flux based on Gauss' Lawe is calculated as: Φ = qin / ε₀ = λl / ε₀
- Electric Field is therefore: E = λ / 2πε₀r
Electric Field due to a Uniformly Charged Infinite Sheet
- σ is defined as surface charge density
- The Gaussian surface is cylindrical
- The total flux is equal to: Фnet = Ф₁ + Ф₂ + Ф₃
- Total flux calculates to: = EA + EA
- Total Flux can be written therefore: Φ = 2EA
- According to Gauss law: Φ = qin / ε₀
- Substituting for surface charge density: 2EA = σA / ε₀
- Electric field calculates to: E = σ / 2ε₀
- It is independent of r
Electric Potential due to a Point Charge
- Potential (V) = Work (W) / charge (q₀)
- Potential: V= kQ/r or V = 1/(4πε₀) * Q/r
- Work done to bring q₀ from infinity to point P: W = ∫p∞ Fex dr
- The derived expression for potential is: W(P→∞) = kQq₀ / r
Electric Potential due to an Electric Dipole
- Electric potential: V = kQ/r
- The net potential: Vnet = V₁ + V₂
- Given the arrangement: Vnet = -kQ/(x+l) + kQ/(x-l)
- After simplification: Vnet = kQ * (2l) / (x²-l²)
- Since p=2lQ the above becomes: Vnet = kP / (x²-l²) or if x>>>l then net becomes kp/x² or V = 1/(4πε₀) * (p / x²)
Potential Energy of an Electric Dipole in a Uniform Electric Field
- The torque (τ) on a dipole in an electric field is: τ = p x E, and (τ = pEsinθ)
- Net work done is: Wel = ∫θ₁^θ₂ τ dθ = ∫θ₁^θ₂ pEsinθ dθ
- Which simplifies to: Wel= pE[cos θ₂ - cos θ₁]
- Change in potential energy (dU) with the equation: dU = -Wel; and U₂-U₁ = -pE(cosθ₂ - cosθ₁)
- Using the equations for U, potential energy can then derived as: U = -p E cos θ and can be expressed as U = - p · E
- Case I: When θ = 90°, τ = pEsin90 = pE and U = - pEcos90 = 0
- Case II: When θ = 0° or 180°: the corresponding values are calculated.
Energy Stored in a Capacitor and Energy Density
- The amount of work done to add charge to a capacitor is stored as electric potential energy
- Capacitance, C = Q/V
- The work needed to add a small amount of charge is dw = dqv
- Performing the integration to find total energy:
- w = ∫dw=∫q/c dq between the limits Q to 0
- This calculated to w = Q²/2C
- To find what the potential energy is stored as, the following equality must hold- U = Q²/2C
- Energy calculations: U= 1/2 (C)V² /C = 1/2CV² = 0
Derivation of Energy Density
- Derivation: U = 1/2 * Q/V * V² which simplifies to U = ½ QV.
- Energy density- Energy/Volume - This calculates to : u = (1/2 CV²) / Ad = 1/2 ε₀ A(Ed)² / Ad
- Cancelations and simplifications leads to: u = ½ ε₀ E²
Drift Velocity
- The average thermal velocity if electrons in a conductor is zero: Vav = 0
- With an electric field (E) applied, an electron experiences a force: F = qE
- Electron acceleration (a) is: a = F/m = -eE/m
- Drift velocity is opposite the direction of electric field
- Average drift velocity of electrons: Vd = -eEτav / m
Drift Velocity in Terms of Current
- In terms of Potential: dv = -Edr, ΔV = El, so, Vd = eV t / ml
- Current defined as the charge (dq) that passes through the conductor per unit time (dt)
- I = dq/dt = nVdaeA, in conventional form I = nVneA
Wheatstone Bridge
- When using Kirchoff's Laws there is a balanced condition
- P, Q, R, and S are resistors
- Under Kirchoff's Rules they are connected in a way that produces the following equation: -PI₁-GIg+R(I-I₁) =0, -IP-IgG+(I-I₁) R=0
- In BCDB, -Q (I-Ig) +s (Ig+I-I₁) +GIg=0
- The bridge is balanced when there is no current flow Ig=0 through the galvanometer; Therefore:- (I-I₁)R = I₁P, I₁Q = S(I-I₁)
- By dividing these equations leads the situation to: P/Q = R/S
Meter Bridge and Wheatstone Bridge
- Whetstone bridge, under balanced conditions, P/Q = R/S
- When the bridge is balanced, the galvanometer shows a null point
- Resistance is then calculated by: R - Sl/(100-l)
Potentiometer
- To determine internal resistance of the test cell
- It follows the expression: E = V × *l₁
- From the equations the length of the wire is x and it can be said: (V ×)/l*l₂ = ER/I+r
- Total resistance, the following derivation of the value can be measured using this equation: E/ER / R+ r = l1*R+r/l₂ x
- Through more math an electronics relationship between these values turns out to be: r/R-l1/l₂ = R /l₁/l ₂-1
Biot-Savart Law
- The Biot-Savart Law is expressed as: dB= (µ₀/4π) * (Idl sinθ)/r²
- where: dB is the magnetic field contribution, µ₀ is the permeability of free space, Idl is the current element, θ is the angle between the current element and the field point, and r is the distance from the current element to the field point
Magnetic Field of a Circular Coil
- Integrate field (B): B = ∫dBsinθ = ∫ (µ₀/4π) * (Idl sinθ)/r²
- Integrate over loop: B = µ₀IR² / 2(R²+a²)^(3/2)
Magnetic Field Inside a Solenoid
- Apply Ampere's Law: ∮ B ⋅ dl = μ₀I
- In the Amperian loop, it follows ∮ B ⋅ dl = B ∫ dl + 0 + 0 + 0 = B l
- In Ampere's Law there are turns (n) per unit length: n = N/h [height]
- After simplification, the result is: B= µ₀nI Tesla
Ampere Law & Toroid
- Ampere's Law: ∮B⋅dl = μ₀Iin
- With the Ampere equation, to find how Many numbers a turn has per unit length requires: n=N/L, n is also represented this way: N / 2πr Because those numbers were involved in our unit path
Helical Path of a Charged Particle in a Magnetic Field
- Force expression: Fm = qvBsin θ
- Set magnetic force equal to centripetal force expression to isolate all motion: qvBsin θ=Mv2/r
- After this find the formula for a radius so that all motion in path is accounted for: R = Mvsinθ Q/B
- Period T, this is equal to: p=vcos θ *T, T = 2πR / vsin θ = 2πm / Q/B
- And thus we finally have our total motion path covered: P pitch = vcos θ * 2πm / Q/B
Force Between Two Parallel Current-Carrying Conductors
- The 2nd wire is placed in B field of 1st wire; So the 2nd wire now experiences force I₂ B₁ l sinθ
- In equation form it now looks like this (F/l) = μ/4π *(2 I₁I₂/D), D = distant point
Magnetic Dipole Moment of an Orbiting Electron
- Angular momentum L = mvr
- Find the ratio of the orbital quantum number in respect to momentum: M/ L = (evr/2) * 1/mvr
- To determine the magnetic dipole, we have:µ =I* A = ev/2r * πr^2 =evr/2
Explanation of the Motional EMF Concept
- An external force, is required to induce a charge by motion
- Using an external force, the magnetic field is now generating motion in an electronic circuit.
- And because work needs to be done to move a charge through space, the equation is in this form: work = q VB
- To find the relationship to the charge to the potential difference, we use these values that give us. BlV
Mutual Inductance of Two Coaxial Solenoids
- With the value found of current going threw the coil, B = µ°NI/ l and magnetic flux Ф°= N1 Ф° = N2 (µ°N1/ LI) A
- Relationship between magnetic movement and inductance with mutual inductance given at: M21, This provides a model for all sorts of circuits that are powered from AC circuits.
Inductor with Voltage Ahead of Current
- Voltage and current (7/2 in phase): V = Vm sin wt
- VL= L di/dt; Vm sin wt = di/dt Reactance
- The function of reactance in the form of the integrated parts used in circuits and is given as : XL= wL (ω = angular speed of the circuit itself)
- And its relationship to voltage and current becomes simplified due to reactance VL=I×XL ; I= Vm/ XL sin (ωt - π/2) Where VLis the inductor its self
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