The cylinder shown has a uniform charge density ρ0 and a total charge of Q. Find b. The knowns are r, ρ0, and Q. The cylinder in the previous problem is given a volume charge density of ρ = ρ0(zz2+b2). If the total charge on the cylinder is Q, find b. The knowns are r, ρ0, and Q.
Another infinite line of charge with charge density λ2 = 10.5 μC/cm parallel to the y-axis is now added at x = 4.15 cm as shown. What is the new value for Ex(P), the x-component of the electric field at point P ? What is the total flux Φ that now passes through the cylindrical surface? Enter a positive number if the net flux leaves the cylinder and a negative number if the net flux enters the cylinder.
Consider a ring of charge in Figure 5 at radius ρ = a and z = 0 with linear charge density ρL, determine the potential V at any point on the z− axis. Figure 5:
A line charge density λ of length L stands perpendicularly on an infinite conducting grounded plane. a) Find the electric field at the ground plane surface E as a function of radius r (reference figure). b) Find the surface charge density on the ground plane. c) Prove that the total charge induced on the plane is −λL d) Determine the potential as a function of r and z for z > > L
A non-conductive rod has a uniform positive charge density +λ, a total charge Q along its right half, a uniform negative charge density −λ, and a total charge −Q along its left half, see Fig. below. (a) What is the electric potential at point A? (b) What is the electric potential at point B?
Q2 (20 pts). A volumetric charge density exists between two spherical shells defined by radii a[m] and b [m], respectively. If b > a and no charge exists anywhere else, determine the total charge Q enclosed between the shells. The charge density is given as ρv = ρ0 Ra[C/m3], where ρ0 is a constant and R is the radial coordinate.
Q5 (5 pts) Electrical charge is distributed uniformly with volume charge density ρ inside a spherical shell of inner radius a and outer radius b. There is a positive point charge Q located at the center. Electric field inside the shell is given by E→ = Cr where C is a positive constant r is the radial distance from the center. (a) Calculate ρ of the spherical shell. (b) Calculate the E field for r > b.
A infinite line of charge has charge density λ = +0.8 μC/m. You correctly use Gauss's law to find the magnitude of the electric field outside the line of charge is E(x) = λ2πε0 x. What is V(d)−V(c), where d = 8 m and c = 3 m ? Be careful to note the order of the potential difference in question. 14104 V −14104 V 29903 V −29903 V 15798 V −15798 V 2996 V −2996 V
3.29 Field due to volume charge density. A short plastic cylinder of length L[m] and diameter L/2[m] has a uniform volume charge density ρv[C/m3] distributed throughout its volume. Calculate the electric field intensity at P1, P2, and P3 (on the axis of the cylinder) shown in Figure 3.46. Figure 3.46
Next two questions: We have a ring with radius R = 3 m with linear charge density λ = 50 μC/m. We have a point charge q = 80 μC that moves along the z-axis through the center of the ring. Point S is zS = 16 meters above the center, and point P is zP = 2 meters above the center. Remember that the electric field along the z-axis above a ring is E→ring = 2πkλRz(z2+R2)3/2 z^Change in potential energy from a ring 1 point Find the change in energy from point S to point P. Figure 1: Figure for Problems 1-2 Change in energy with opposite path 1 point Find the change in potential energy from point P to point S.
Monday Homework Problem 2.8 Most of you found a linear charge density for the golf tube of around −1.0× 10−7 Cm by modelling the tube as an infinite straight uniform line charge. The golf tube has a radius of about 2.0 cm. What is the magnitude of the electric field at the surface of the golf tube if it is modelled as an infinite line of charge? Select One of the Following: (a) 9.0×104 NC (b) 4.5×104 NC (c) 1.1×106 NC (d) 2.2×106 NC (e) 6.0×103 NC
A surface charge density ρs = ρs0 C/m2 lies on surface defined by a ≤ R ≤ b, 0 ≤ θ ≤ π/2 and ϕ = 0 as shown in Figure 1. Calculate the Electric field E→ at the point (0, 0, 0).
An insulating solid sphere of radius R has a volume charge density ρ = ρ0. Calculate the electric field at a point outside and inside the sphere. A thin cylindrical shell of radius R and height h is placed in the xy plane with its base at the origin. The cylinder has a uniform σ charge density, find the electric field at point P on its z axis (Use E = k∫dqr2 ).
We have an infinitely long wire, again with linear charge density λ = 10 μC/m. We have a point charge, again with q = 75 μC. Point S is zS = 10 m away from the wire on the z-axis, point P is zP = 5 m away from the wire on the z-axis, and point T is yT = 5 m away from the wire on the y-axis. Remember that the electric field at a distance r away from a wire is E→wire = 2 kλrr^ Figure 2: Figure for Problems 3-5 3. Potential change with radial path 1 point What is the energy change from point S to point P ? 4. Potential change with angular path 1 point What is the change in energy from point P to point T along the quarter-circle path? 5. Potential change with square path 1 point What is the change in energy from point P to point T along the square path? (Next two questions) We have an infinitely long wire of linear charge density 1 nC/m. You throw a ball with charge Q = −8 C and mass m = 2 kg away from the wire with initial velocity vi = 12 m/s. The ball is initially 4 meters away from the wire. 6. Change in potential 1 point After some time the ball's velocity decreases to 6 m/s, half its initial speed. What is the magnitude of the change in potential, in volts, from the initial position to this position of the ball? 7. Maximum distance 2 points What is the maximum distance the ball gets from the wire?
Q7. (Electric field due to continuous charge distribution) A semi-circular ring lying in the y−z plane has a charge density ρl = ρ0 cosθ' coul /m, where θ′ is the angle measured from the z-axis as shown in the figure. (a) Find E for points (x, 0, 0) along the x-axis. (b) Show that for x ≫ R, the electric field is like that of a dipole, i. e. , depends on the cube (x3) of the displacement from the charge distribution.
For the given positive charge distribution with uniform volume charge density ρ0, determine the electric field vector at point A. Assume the initial blue part is a non conducting sphere and you have removed the portion of the sphere in white (spherical cavity)
A total charge Q is uniformly distributed, with surface charge density σ, over a very thin disk of radius R. The electric field at a distance d along the disk axis is given by E→ = σ2ϵ0[1−dd2+R2]n^ where n^ is a normal unit vector perpendicular to the disk. What is the best approximation for the electric field magnitude E at large distances from the disk? a) σ2ϵ0 b) QRϵ0 d2 c) σR4ϵ0 d d) σϵ0(R2 d)2 e) none of the above
Consider a uniformly distributed spherical charge density of −ρv with radius b. Find the electric field intensity for 0 < r < b and b < r. a. Use Gauss's law b. Use electric potential
A uniformly charged wire of linear charge density λ is bent into the shape shown in figure. Find electric potential at a point O in terms of R, L and λ.
A series of line charges each with constant charge density λ are arranged to create a square with side length s as shown below. (a) Find the electric field a height z above the center of the square. (Hint: consider each side of the square independently, and add all of the electric fields together. ) (b) Suppose s = 1 m, λ = 2.4 μC/m, and z = . 5 m. What is the force on an electron (q = −1.6×10−19 C and m = 9.1×10−31 kg ) placed at this point? What is its acceleration?
A very long line charge having a charge density λ is surrounded by a conducting cylindrical shell with inner radius ri = 4.05 cm and outer radius r0 = 7 cm as shown in the figure. System is in electrostatic equilibrium condition. What would be the electric potential difference ΔV = V(r = 1.47 cm)−V(r = 6.2 cm) ? Provide your answer in terms of λk with 2 significant figures.
Suppose that the charge density of the spherical charge distribution shown in the figures below is ρ(r) = ρ0r/R for r ≤ R and zero for r > R. Obtain expressions for the electric field both inside and outside the distribution. (Use any variable or symbol stated above along with the following as necessary: ε0. ) (a) (b) inside E→ = r^ outside E→ =
A non-conducting hollow cylinder has a uniform volume charge density of 1.2 c/m3. The inner radius of the cylinder is 0.25 m, while the outer radius is 0.35 m. Calculate the magnitude of the electric field at r = 0.32 m. A 2.7×109 N/C B 1.3×1010 N/C C 4.3×109 N/C D 2.2×1010 N/C E 8.5×109 N/C
A solid cylinder, with radius a, contains a uniform charge density ρ. The cylinder is coaxially aligned (see figure 2) inside a cylindrical shell, with radius b, carrying a charge per unit area σ. If σ = −a2ρ/2b, and ρ is positive, use Gauss's law to find the electric field as a function Figure 2. Problem 2. of distance from the central axis in the following regions. (a) r < a (b) a < r < b (c) r > b.
The charge density on a disk of radius R = 11.0 cm is given by σ = ar, with a = 1.38 μC/m3 and r measured radially outward from the origin (see figure below). What is the electric potential at point A, a distance of 38.0 cm above the disk? Hint: You will need to integrate the nonuniform charge density to find the electric potential. You will find a table of integrals helpful for performing the integration.
A thin wire with uniform charge density and total charge Q is bent into a 1/4 circle or radius R as shown. The position P sits at the center of the circle. The z-axis sticks up out of the page Find the potential, V(0, 0, z), i. e. as a function of the z-coordinate for x = 0, y = 0, Find the z-component of the electric field, Ez, where z is the direction out of the page by taking the appropriate partial derivative of V. Compute E→ directly by integrating E→ = ∫dE→ For this problem, consider splitting the arc up into infinitesimal segments of angle dθ, find the contribution from each segment, and integrate the result to find the total answer. All vector quantities (e. g. E→ but not V or Ez ) must be expressed in components using unit vector notation (e. g. v→ = vxi^+vyj^+vzk^
An insulating solid sphere of radius R has a non-uniform charge distribution with volume charge density ρv = ρ0r/R where ρ0 is a constant and r is the distance from the center of the sphere to the point of interest. What is the magnitude of the electric field inside the sphere (for r < R )? [NOTE: To compute the enclosed charge, consider differential spherical shell elements of radius s and thickness ds so that the differential volume dv = 4πs2 ds and the differential charge dq = ρvdv ] Er = ρ0 R2 ε0 r2 Er = ρ0 R24 ε0 r2 Er = ρ0 r22 ε0 R Er = ρ0 r24 ε0 R
The electric charge density is distributed symmetrically in a cylinder infinitely long in the z direction. The charge density is given by the expression: ρe(ρ) = {ρ0(ρb)2, ρ ≤ b0, ρ > b where ρ is the cylindrical coordinate, ρ0 is a constant and b is the radius of the cylinder. a) Find expressions for the electric field in the region ρ < b and the region ρ > b. b) If a grounded metallic shell is added at ρ = a(a > b) such that the electric field E→ = 0 for ρ > a. Calculate the electric surface charge density on the shell.
A thin semi-circle of uniform charge density λ has a radius R and subtends an angle ϕ. There is a total charge +Q on the entire arc, and the arc is positioned in the xy-plane. a) Find the dq and the r^ needed to use the equation dE→ = krdqr^. How many "dimensions" are you integrating over? What are the integration bounds for this situation? b) Now use a symmetry to simplify the vector expression that you got in the last part and use it to evaluate the total electric field at the point P. c) What happens to the electric field as ϕ → 2π ? Does that make intuitive sense?
An infinite line of charge with charge density λ1 = −4.8 μC/cm is aligned with the y-axis as shown. What is Ex(P), the value of the x-component of the electric field produced by by the line of charge at point P which is located at (x, y) = (a, 0), where a = 7.2 cm ? N/C 2) What is Ey(P), the value of the y-component of the electric field produced by by the line of charge at point P which is located at (x, y) = (a, 0), where a = 7.2 cm ? N/C Submit
A very long wire that has a linear charge density of +5.0 C/m lies along the y axis. A second wire, parallel to the first, has an x position of +1.0 m meters. It has a charge density of −2.0 C/m. At what points on the x axis is the electric field strength equal to zero?
Two lines of charge with uniform, linear charge density, λ are located on the z and y axes as shown. Both have length ℓ and are located a distance a, and b, respectively from the origin. Find the total Electric field at the origin.
3.25 Electric field due to surface charge density. A disk of radius a[m] is charged with a uniform surface charge density ρs = ρ0[C/m2]. Calculate the electric field intensity at a distance h from the center of the disk on the axis (see Figure 3.44). Figure 3.44
3.25 Electric field due to surface charge density. A disk of radius a[m] is charged with a uniform surface charge density ρs = ρ0[C/m2]. Calculate the electric field intensity at a distance h from the center of the disk on the axis (see Figure 3.44). Figure 3.44
An infinite line of charge with charge density λ1 = −3.5 μC/cm is aligned with the y-axis as shown. What is Ex(P), the value of the x-component of the electric field produced by by the line of charge at point P which is located at (x, y) = (a, 0), where a = 8.3 cm ? N/C
The initial infinite line of charge is now moved so that it is parallel to the y-axis at x = −4.15 cm. What is the new value for Ex(P), the x-component of the electric field at point P ? Submit 7) What is the total flux Φ that now passes through the cylindrical surface? Enter a positive number if the net flux leaves the cylinder and a negative number if the net flux enters the cylnder.
1. The volume charge density ρv = ρ0⋅e−|x|−|y|−|z| exits over all free space. Calculate the total charge present . 2. A line charge with charge density equal to 10 nC/m is at x = 4, z = 3. A sheet of charge with surface charge density equal to −1 nC/m is at the xy - plane as shown in Figure 1. What is thw electric field at P(2, 3, 5) ? Figure 1:
A sphere with a uniform charge density ρ and radius R has a cavity of radius R/2 as shown in the figure below. Find the electric field at a point (x, y, z) outside the sphere. The center of the sphere is at (0, 0, 0), and the center of the cavity is at (R/2, 0, 0).
A semi-infinite slab of charge with charge density ρ extends from x = −d to x = d. Find a vector expression for the electric field everywhere i. e. in the regions (i) x < −d, (ii) −d < x < d, (iii) x > d.
A ball of radius s carries a charge density proportional to the distance from the origin. In other words, the charge density function is given by kx2+y2+z2. Find the total charge in the ball in terms of k and s. Hint: the integral of ∗ density over a region is the total ∗ in the region, no matter what ∗ is.
A charged sphere of radius a has a spherically symmetric charge density ρ(r) = kr that varies linearly with the distance r from the center O. The total charge of the sphere is Q. This charged sphere is surrounded by a grounded conducting sphere of radius b > a that is also centered at O, as shown in the Figure. (1) Find the electric field E(r) everywhere in space. (2) Plot the magnitude of the electrostatic field as a function of the distance r from the center O. (3) Find the electrostatic potential everywhere in space. (4) Plot the magnitude of the electrostatic potential as a function of the distance r from the center O.
A coaxial cable has an inner conductor of radius a and an outer conductor of radius b, as shown. The inner conductor has a uniform surface charge density ρs0[C/m2]. The outer conductor has a uniform surface charge density −ρs0 [C/m2] Find the electric field vector in the three regions 0 < ρ < a, a < ρ < b, b < ρ.
A sphere of radius R1 = 0.320 m and uniform charge density 37.5 μC/m3 lies at the center of a neutral, spherical, conducting shell of inner and outer radii R2 = 0.588 m and R3 = 0.720 m, respectively. Find the surface charge density on the inner and outer surfaces of the shell. inner surface charge density: μC/m2 outer surface charge density: μC/m2
An infinite line of charge (surface charge density ρs ) is oriented along the z-axis. It is located a distance d above a grounded, conducting half-plane oriented in the x−z plane and a distance d away from a grounded, conducting half-plane oriented in the y−z plane, as shown below. Consider the halfplanes to be infinite. a) Draw the locations, magnitudes, and polarities of the image lines of charge with respect to each of the two grounded conductors. b) Find the electric field at an arbitrary point P = (x, y, 0).
The figure below shows two lines of charge in cross-section coming out of the page. Each line of charge has a linear charge density of 7.90 nC/cm and they are separated by a distance d = 20.8 cm. What is the magnitude of the electric field at the dot, a point y = 5.70 cm above the midpoint between the lines? Electric field magnitude: ◻
Suppose a ring-shaped thin metal sheet charged with a uniform surface charge density ρs and a finite wire of length 2L with a uniform line charge density ρl. The sheet has an inner and outer radius given by a and b and the center of the wire coincides with the center of the ring, as shown in the illustration. (1) Use Coulomb's Law to calculate the electric field at z > L and z < −L. (2) Under what conditions, can you make the electric field at z equal to zero for z > L and z < −L ? (3) Apply the result of (1) to calculate the electric field of an infinite surface charge.
The linear charge density on a ring of radius a is given by ρℓ = qa(cosϕ−sin2ϕ). Show that: (a) qT( monopole moment ) = 0 (b) p→ (dipole moment ) = (πqa, 0, 0) (c) Qij (quadrupole moment) = 32πqa2(0 −1 0 −1 0 0 0 0 0)
Two large charged plates of charge density ±44 μC/m2 face each other at a separation of 5 mm. Choose coordinate axes so that both plates are parallel to the xy plane, with the negatively charged plate located at z = 0 and the positively charged plate at z = +5 mm. Define potential so that potential at z = 0 is zero (V(z = 0) = 0). Hint a. Find the electric potential at following values of z : 。 potential at z = −5 mm : V(z = −5 mm) = V. ० potential at z = +3.2 mm : V(z = +3.2 mm) = 16000 ० potential at z = +5 mm : V(z = +5 mm) = 25000 V ० potential at z = +5.8 mm : V(z = +5.8 mm) = 29000×V. b. An electron is released from rest at the negative plate. With what speed will it strike the positive plate? The electron will strike the positive plate with speed of m/s. (Use "E" notation to enter your answer in scientific notation. For example, to enter 3.14×1012, enter "3.14 E12". )
Consider a uniformly-charged radius- R insulating sphere with volume charge density ρ with an off-center spherical cavity of radius R/2 as shown to the right. What is the electric potential at point P assuming that the potential is zero infinitely away from the sphere? Hint: Consider the superposition of two spheres, one with a negative charge density. A. V = ρR24ϵ0 B. V = 7ρR224ϵ0 C. V = ρR23ϵ0 D. V = 0 E. None of the above.