Podcast
Questions and Answers
What is the potential on the axis of a uniformly charged disk?
What is the potential on the axis of a uniformly charged disk?
- $2k\sigma\pi|z| [\sqrt{1 + \frac{R^2}{z^2}} - 1]$
- $kQ/|z|^2$ (correct)
- $kQ^2/|z|$
- $2k\sigma\pi|z|$
For $|z| >> R$, which potential function should $V$ approach?
For $|z| >> R$, which potential function should $V$ approach?
- Infinity, due to the large distance.
- Zero, since the distance is very large.
- The potential function of a point charge $Q$ at the origin. (correct)
- The potential function of a uniformly charged disk.
What approximation is used for $|z| >> R$?
What approximation is used for $|z| >> R$?
- Fourier transform
- Binomial expansion (correct)
- Taylor series
- Laplace transform
What is the simplified potential $V(z)$ when $|z| >> R$?
What is the simplified potential $V(z)$ when $|z| >> R$?
What does $Q$ represent in the context of the potential of a charged disk?
What does $Q$ represent in the context of the potential of a charged disk?
What is the purpose of using approximations like the binomial expansion when calculating electric potential?
What is the purpose of using approximations like the binomial expansion when calculating electric potential?
In the equation $V(z) = 2k\sigma\pi|z| [\sqrt{1 + \frac{R^2}{z^2}} - 1]$, what does $\sigma$ represent?
In the equation $V(z) = 2k\sigma\pi|z| [\sqrt{1 + \frac{R^2}{z^2}} - 1]$, what does $\sigma$ represent?
What happens to the electric potential as $|z|$ becomes much larger than $R$?
What happens to the electric potential as $|z|$ becomes much larger than $R$?
What is the area element $dA$ of the ring in Figure (8)?
What is the area element $dA$ of the ring in Figure (8)?
What is the expression for the charge $dq$ of the ring?
What is the expression for the charge $dq$ of the ring?
What does $\sigma$ represent in the context of the charged ring?
What does $\sigma$ represent in the context of the charged ring?
What is the formula for the surface charge density, $\sigma$, in terms of total charge $Q$ and radius $R$?
What is the formula for the surface charge density, $\sigma$, in terms of total charge $Q$ and radius $R$?
What is the potential $dV$ at point $P$ due to the charged ring?
What is the potential $dV$ at point $P$ due to the charged ring?
Over what range of $a$ do we integrate in order to find the total potential due to the charged disk?
Over what range of $a$ do we integrate in order to find the total potential due to the charged disk?
What is the form of the integral used to find the total potential $V$?
What is the form of the integral used to find the total potential $V$?
What are the limits of integration for $u$ when integrating with respect to $u$?
What are the limits of integration for $u$ when integrating with respect to $u$?
If a charge distribution possesses spherical symmetry, the electric field is a function of what?
If a charge distribution possesses spherical symmetry, the electric field is a function of what?
What is the formula to obtain the electric field $\vec{E}$ if $V(r)$ is known?
What is the formula to obtain the electric field $\vec{E}$ if $V(r)$ is known?
What is the electric potential $V(r)$ due to a point charge $q$?
What is the electric potential $V(r)$ due to a point charge $q$?
What is the electric field $\vec{E}(r)$ due to a point charge q?
What is the electric field $\vec{E}(r)$ due to a point charge q?
If the electric potential is $V(x) = 100 , V - (25 , V/m)x$, what is the $x$-component of the electric field, $E_x$?
If the electric potential is $V(x) = 100 , V - (25 , V/m)x$, what is the $x$-component of the electric field, $E_x$?
If the electric potential does not vary with $y$ or $z$, what are the values of $E_y$ and $E_z$?
If the electric potential does not vary with $y$ or $z$, what are the values of $E_y$ and $E_z$?
What is $dV$ equal to in a spherically symmetrical charge distribution?
What is $dV$ equal to in a spherically symmetrical charge distribution?
The electric field, $\vec{E}$, is related to the potential $V$ by which mathematical operation?
The electric field, $\vec{E}$, is related to the potential $V$ by which mathematical operation?
What does $dV$ represent in the equation $dV = -\vec{E} \cdot d\vec{s}$?
What does $dV$ represent in the equation $dV = -\vec{E} \cdot d\vec{s}$?
In Cartesian coordinates, what is the vector form of the differential displacement $d\vec{s}$?
In Cartesian coordinates, what is the vector form of the differential displacement $d\vec{s}$?
Given $V = V(x, y, z)$, which expression represents the total differential $dV$?
Given $V = V(x, y, z)$, which expression represents the total differential $dV$?
What does the negative sign in the equation $\vec{E} = -(\frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k})$ indicate?
What does the negative sign in the equation $\vec{E} = -(\frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k})$ indicate?
What is the relationship between the electric field $\vec{E}$ and the electric potential $V$?
What is the relationship between the electric field $\vec{E}$ and the electric potential $V$?
In the equation $dV = E_x dx + E_y dy + E_z dz$, what does $E_x$ represent?
In the equation $dV = E_x dx + E_y dy + E_z dz$, what does $E_x$ represent?
What does $\frac{\partial V}{\partial x}$ represent?
What does $\frac{\partial V}{\partial x}$ represent?
If the electric potential $V$ increases as a positive charge moves along the x-direction, what can be said about the x-component of the electric field, $E_x$?
If the electric potential $V$ increases as a positive charge moves along the x-direction, what can be said about the x-component of the electric field, $E_x$?
What is the value of the potential $V$ everywhere on an equipotential surface?
What is the value of the potential $V$ everywhere on an equipotential surface?
If a test charge on an equipotential surface is given a small displacement $dl$ parallel to the surface, what is the value of $dV$?
If a test charge on an equipotential surface is given a small displacement $dl$ parallel to the surface, what is the value of $dV$?
What is the relationship between electric field lines and equipotential surfaces?
What is the relationship between electric field lines and equipotential surfaces?
For a point charge, what shape do equipotential surfaces form?
For a point charge, what shape do equipotential surfaces form?
For a constant electric field, what shape do equipotential surfaces form?
For a constant electric field, what shape do equipotential surfaces form?
What is the direction of the electric field with respect to equipotential surfaces?
What is the direction of the electric field with respect to equipotential surfaces?
What does $E \cdot dl = 0$ imply when $dl$ is parallel to the surface?
What does $E \cdot dl = 0$ imply when $dl$ is parallel to the surface?
Which of the following is true about equipotential surfaces?
Which of the following is true about equipotential surfaces?
What does $V(x)$ represent in the context of the electric potential due to a charged ring?
What does $V(x)$ represent in the context of the electric potential due to a charged ring?
What component(s) of the electric field exists along the symmetry axis of a charged ring?
What component(s) of the electric field exists along the symmetry axis of a charged ring?
How is the electric field component $E_x$ related to the electric potential $V$?
How is the electric field component $E_x$ related to the electric potential $V$?
In the equation for $E_x(x)$ for a charged ring, what does 'a' represent?
In the equation for $E_x(x)$ for a charged ring, what does 'a' represent?
If the electric potential $V(x, y, z) = A x^2 y^2 + B x y z$, how is $E_x$ calculated?
If the electric potential $V(x, y, z) = A x^2 y^2 + B x y z$, how is $E_x$ calculated?
Given $V(x, y, z) = A x^2 y^2 + B x y z$, what is the expression for $E_z$?
Given $V(x, y, z) = A x^2 y^2 + B x y z$, what is the expression for $E_z$?
What are curves characterized by constant $V(x, y)$ called?
What are curves characterized by constant $V(x, y)$ called?
What does $\epsilon_0$ represent in the equation for electric potential?
What does $\epsilon_0$ represent in the equation for electric potential?
What is the electric field a measure of?
What is the electric field a measure of?
What is the relationship between electric potential and electric field?
What is the relationship between electric potential and electric field?
Flashcards
V(z) for a charged disk
V(z) for a charged disk
Potential on the axis of a uniformly charged disk.
V(z) when |z| >> R
V(z) when |z| >> R
For distances much larger than the disk's radius, potential approaches that of a point charge.
Binomial expansion
Binomial expansion
Approximation used when |z| is much larger than R.
Approximated V(z) for |z| >> R
Approximated V(z) for |z| >> R
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σπR^2
σπR^2
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k
k
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Electric Field from Potential
Electric Field from Potential
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Eqn. (1)
Eqn. (1)
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dA (Disk Area)
dA (Disk Area)
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Surface Charge Density (σ)
Surface Charge Density (σ)
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Total Potential V
Total Potential V
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V(z)
V(z)
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kdq / √(z² + a²)
kdq / √(z² + a²)
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Potential of charged disk
Potential of charged disk
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dV = -E · ds
dV = -E · ds
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E in Cartesian Coordinates
E in Cartesian Coordinates
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ds in Cartesian Coordinates
ds in Cartesian Coordinates
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dV in Cartesian coordinates
dV in Cartesian coordinates
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dV and Partial Derivatives
dV and Partial Derivatives
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Electric Field Components
Electric Field Components
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E as Gradient of V
E as Gradient of V
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Negative Sign Significance
Negative Sign Significance
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Spherical Symmetry & Electric Field
Spherical Symmetry & Electric Field
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dV in Spherical Symmetry
dV in Spherical Symmetry
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Electric Field from Potential (Spherical)
Electric Field from Potential (Spherical)
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Potential of a Point Charge
Potential of a Point Charge
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Electric Field of a Point Charge (from V)
Electric Field of a Point Charge (from V)
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Ex from V(x)
Ex from V(x)
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Potential Independent of y, z
Potential Independent of y, z
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E Field from V(x) Example
E Field from V(x) Example
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Electric Field (Ex) Formula
Electric Field (Ex) Formula
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Finding Ex
Finding Ex
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Finding Ey
Finding Ey
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Finding Ez
Finding Ez
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Equipotential Surface
Equipotential Surface
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Equipotential Curves
Equipotential Curves
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Electric Field Direction
Electric Field Direction
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Electric Field of a Ring
Electric Field of a Ring
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Electric Potential of a Ring
Electric Potential of a Ring
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Electric Field Vector
Electric Field Vector
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dV on Equipotential Surface
dV on Equipotential Surface
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E-field and Equipotential
E-field and Equipotential
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Equipotential of Point Charge
Equipotential of Point Charge
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Equipotential in Uniform E-field
Equipotential in Uniform E-field
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Direction of E-field
Direction of E-field
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Conductor Potential
Conductor Potential
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Force and Equipotential
Force and Equipotential
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Study Notes
- The potential at a point P with continuous charge distribution is the sum of individual differential charge elements dq contributions
- With infinity as a zero-potential reference the electric potential at P due to dq is dV = (1 / 4πε₀) * (dq / r)
- The total potential over all differential elements is then V = ∫dV = ∫(1 / 4πε₀) * (dq / r)
Electric Potential on the Axis of a Charged Ring
- For a uniformly charged ring of radius a and charge Q in the z = 0 plane, centered at the origin, the distance from a charge element dq to field point P on the ring's axis is r = √(z² + a²)
- The potential at a point due to the ring is obtained by integrating: V = ∫(k * dq / r) = (k / r) ∫dq = kQ / r = kQ / √(z² + a²)
- When |z| is much greater than a, the potential approaches kQ / |z|, similar to a point charge Q at the origin
Electric potential V for a Charged Disk
- Potential on the axis of a disk with radius R and total charge Q distributed uniformly is found by treating the disk as rings
- A ring with radius a and thickness da in Fig. (8) has an area of dA = 2πa da
- The charge of the ring is dq = σdA = σ2πa da, where σ = Q / πR² (surface charge density)
- Potential at point P due to ring charge is dV = kdq / √(z² + a²)
- Integrate from a = 0 to a = R to find the total potential V = ∫₀ᑅ (kσ2πa da) / √(z² + a²) = kσπ[√(z² + R²) - |z|]
- Rearranged, V(z) = 2kσπ|z| * [√(1 + R² / z²) - 1]
- The potential on the axis of a uniformly charged disk is V(z) = 2kσπ|z| * [√(1 + R² / z²) - 1]
Electric field from Electric Potential
- The relation between E and V for two points with small distance ds apart is dV = -E · ds
- In Cartesian coordinates: E = Exî + Eyĵ + Ezk and ds = dx î + dy ĵ + dz k
- Therefore: dV = -(Exî + Eyĵ + Ezk) · (dx î + dy ĵ + dz k) which simplifies to dV = Exdx + Eydy + Ezdz
- Also: dV = (∂V/∂x)dx + (∂V/∂y)dy + (∂V/∂z)dz
- Which means: Ex = -∂V/∂x, Ey = -∂V/∂y, Ez = -∂V/∂z
- The electric field can be expressed: E = Exî + Eyĵ + Ezk = -(∂V/∂x)î - (∂V/∂y)ĵ - (∂V/∂z)k
- Mathematically think of Electric Field as the negative gradient of the electric potential V
- Physically, when V increases as a positive charge moves along x, there is a non-vanishing component of E in the x opposite direction (-Ex ≠ 0)
- For charge distribution with spherical symmetry, E is a function of radial distance r: E = Eᵣr̂
- In this case, dV = -Eᵣdr, and if V(r) known, E = Eᵣr̂ = -(dV/dr)r̂
- Example: electric potential due to point charge q is V(r) = 1/(4πε₀) * (q/r), and E = 1/(4πε₀) * (q/r²)r̂
Examples
- For a potential function V(x) = 100 V - (25 V/m)x, because V only depends on x, Ex = -dV(x)/dx = 25 V/m, and Ey = Ez = 0
- For a ring with radius a and charge Q, potential at point P on the ring axis a distance x is V(x) = 1/(4πε₀) * (Q/√(x² + a²))
- The x-component of the electric field is Ex = Qx / [4πε₀(x² + a²)³/²]
- Electric potential due to a charge distribution can be written as V(x, y, z) = Ax²y² + Bxyz where A and B are constants
- Electric field components are found by taking partial derivatives: Ex = -(2Axy² + Byz), Ey = -(2Ax²y + Bxz) and Ez = -Bxy
- The electric field: E(x, y, z) = -(2Axy² - Byz)î - (2Ax²y + Bxz)ĵ - (Bxy)k
Equipotential Surfaces and Field Lines
- A system in 2D with electric potential V(x, y), equipotential curves have constant V(x, y)
- In 3D, equipotential surfaces are described by V(x, y, z) = constant
- The potential V is constant on an equipotential surface, and for a small displacement dl parallel: dV = -E · dl = 0
- E must be zero or perpendicular to every dl parallel to the surface, meaning E is normal to the surface
- Electric field lines are normal to equipotential ones they intersect
- The field lines in the plane of the charges are represented by red lines, intersections by blue lines
- Equipotential surfaces are 3D, at each crossing of equipotential/field line, the two are perpendicular
Equipotential properties
- Electric field lines are perpendicular pointing from high to low potentials
- Equipotential surfaces from point charge form concentric spheres; constant electric field forms planes perpendicular
- The tangential component of Electric field along the equipotential surface is zero, no work is done while moving charge
- No work needed for a particle along equipotential surface
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Description
Explores the electric potential on the axis of a uniformly charged disk, focusing on approximations for distances much larger than the disk's radius. It covers the potential function's behavior, the use of binomial expansion, and the meanings of key variables like surface charge density.
