The figure below shows a finite line charge with linear charge density of λ and total length L. The point P shown is a distance s away from its end.
Please calculate a formula for the electric field at point P, in terms of λ, L and s.
Consider a small charge element, dt at K. Where KO = t
Angle OPK = θ
The electric field due to this element is:
dE = (1/4πε) [ λdt / (S2 + t2 ) ] cosθ i - (1/4πε) [ λdt / (S2 + t2 ) ] sinθ j
Note : the component along Y-axis is negative, since y-axis is upward.
cosθ = S / (S2 + t2)1/2 , sinθ = t / (S2 + t2)1/2
So, t =
S tanθ => dt = S sec2θ
dθ
dE = (λ/4πε) [S sec2θ / (S secθ)2] cosθ dθ i - (λ/4πε) [ Ssec2θ / (Ssecθ)2] sinθ dθ j
= (λ/4πε) (1/S) cosθ dθ i - (λ/4πε) (1/S) sinθ dθ j
Where i and j are unit vectors along X and Y axis respectively.
Integrating: θ varies from 0 to tan-1 (L/S)
Note: tan-1(L/S) = sin-1 [ L / (L2 + S2)1/2 ] = cos-1 [ S / (L2 + S2)1/2 ]
( Just construct a triangle and verify)
in this image θ = tan-1(L/S)
E = (λ/4πε) (1/S) [ L / (L2 + S2)1/2 ] i - (λ/4πε) (1/S) [ - S / (L2 + S2)1/2 + 1 ] j
= (λ/4πε) (1/S) [ L / (L2 + S2)1/2 ] i - (λ/4πε) (1/S) [ 1 - S / (L2 + S2)1/2 ] j
λ = 7 μ C/m, L = 4 m, S = 3 m
E = 7 x 10-6 x 9 x 109 x (1/3) x [ 4 / 5] i - 7 x 10-6 x 9 x 109 x (1/3) x [ 1 - 3/5] j
= 16.8 x 103i - 8.4 x 103j N/C
Ex = 16.8 K N/C
Ey = - 8.4 K N/C
The figure shows a uniformly charged thin rod of length L = 0.531 m that has total charge Q = 5.49 mC. What is the magnitude of the electrostatic force acting on an electron positioned on the axis of the rod at a distance d = 0.401 m from the midpoint of the rod?
The figure shows a uniformly charged thin rod of length L = 0.531 m that has total charge Q = 5.49 mC. What is the magnitude of the electrostatic force acting on an electron positioned on the axis of the rod at a distance d = 0.401 m from the midpoint of the rod?
The figure below shows a finite line charge with linear charge density of λ and total...
30 An infinite line of charge with linear density λ,--S6pcim is positioned along the axis of a thick conducting shell of inner radius a 3.4 cm and outer radius b-54 cm and infinite length. The conducting shell is uniformly charged with a linear charge density A 2 3.5 uC/m 1) What is EXP), the electric field at point P, located at (x,y)卟7.6cm, 0cm) ? NIC Submit 2) What is Ey/P), the electric field at point P, located at (xy)-(-7.6 cm,...
Three line charges of length L and charge density λ are placed parallel to each other, separated by a distance h, as shown in the figure below. Calculate the electric field E at point P, a distance d away from the center line charge. Hint: Find the electric field at the point P for the center line charge and from that, use superposition for the others. l h P d
A finite line of charge with linear charge density A 3.35 x10 Clm, and length L-0.654 m is located along the x axis (from xa 0 to x = L). A point charge of q =-6.22 x 10' C is located at the point xo 1.56 m, yo 4.50 m. Find the electric field (magnitude and direction as measured from the +x axis) at the point P which is located along the x axis at XP-11.7 m. The Coulomb force...
A finite line of positive charge(with linear density lambda) is centered at the origin along on the x-axis from Beginning with dE and integrating the electric field E at a point h on the y-axis is given by:
LC-S A line of length I. with conastant linear charge density +λ(C/m) lies on the x axis. A port P lies on the perpendicular bisector of the line (the y axis). a) Show the details of the symmetry argument (pick dq's and show their dE's) to explairn which of the two components (x or y) of the net field at point P are equal to zero due to the entire line of charge. Use the figure in your explanation. Explicitly...
An infinitely long line of charge has a linear charge density λ, in units of C/m. (a) (3 pts.) Describe the shape Gaussian surface you would use for this charge configuration and the electric flux for this surface. Do all of the parts of this Gaussian surface have a nonzero electric flux? Explain. (b) (3 pts.) Derive an expression for the electric field in terms of the linear charge density λ. (c) (4 pts.) Briefly show how you would find...
30 Line 1 An infinite line of charge with linear density 6.4pC/m is positioned along the axis of a thick conducting shell of inner radius a . 2.8 cm and outer radius b-4.6 cm and infinite length. The conducting shell is uniformly charged with a linear charge density A 2-4.4 HC/m 1) What is E(P), the electric field at point P, located at (x,y) (-10.6 cm, 0 cm)? N/C Submit 2) What is EyIP), the electric field at point P,...
(Figure 1)A charged wire of negligible thickness has length 2L units and has a linear charge density λ. Consider the electric field E at the point P, a distance d above the midpoint of the wire. gure 1 of 2L What is the magnitude E of the electric field at point P? Throughout this part, express your answers in terms of the constant k, defined by k L- Express your answer in terms of L, X, d, and k.
Figure 23-55 shows, in cross section, three infinitely large nonconducting sheets on which charge is uniformly spread. The surface charge densities are σ1 = 2.29 µC/m2, σ2 = 3.30 µC/m2, and σ3 = -3.82 µC/m2, and distance L = 1.19 cm. In N/C, what are the (a) x and (b) y components of the net electric field at point P? Figure 23-55 shows, in cross section, three infinitely large nonconducting sheets on which charge is uniformly spread. The surface charge...
A uniform line charge that has a linear charge density λ = 4.1 nC/m is on the x axis between x = 0 to x-5.0 m (a) what is its total charge? (b) Find the electric field on the x axis at x = 6 m 1 23e 10 × N/C ci the electric field on the x axis at x 8.0 m N/C (d) Find the electric field on the x axis at x 300 m N/C (e)Estimate the...