Use Gauss's law (OE•dÃ ) to show that the electric field a distance r away from the axis enc of an infinitely long straight wire with a uniform charge per length 2 (shown below) is given by Ē = 2arɛ, show your steps and include a clear diagram showing the vectors E and dA.

Use Gauss's law (OE•dÃ ) to show that the electric field a distance r away from the axis enc of an infinitely long straight wire with a uniform charge per length 2 (shown below) is given by Ē = 2arɛ, show your steps and include a clear diagram showing the vectors E and dA.

University Physics Volume 2

18th Edition

ISBN:9781938168161

Author:OpenStax

Publisher:OpenStax

Chapter6: Gauss's Law

Section: Chapter Questions

Problem 84AP: A total charge Q is distributed uniformly throughout a spherical volume that is centered at o1 and...

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Consider a thin plastic rod bent into an arc of radius Rand angle a (see figure below). The rod carries a uniformly distributed negative charge Q -Q A IR What are the components and E, of the electric field at the origin? Follow the standard four steps. (a) Use a diagram to explain how you will cut up the charged rod, and draw the AE contributed by a representative piece. (b) Express algebraically the contribution each piece makes to the and y components of the electric field. Be sure to show your integration variable and its origin on your drawing. (Use the following as necessary: Q, R, cx, 0, A0, and EQ-) ΔΕ, = - TE aR² AB=(2 Lower limit= 0 ✓ e aR² Upper limit= a X cos(0)40 x (c) Write the summation as an integral, and simplify the integral as much as possible. State explicitly the range of your integration variable. sin (0)40 x Evaluate the integral. (Use the following as necessary: Q, R, a, and E.) Edit
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Figure 1 shows four situations in which four very long rods extend into and out of the page (we see only their cross sections). The value below each cross section gives that particular rod’s uniform charge density in microcoulombs per meter. The rods are separated by either d or 2d as drawn, and a central point is shown midway between the inner rods. Rank the situations according to the magnitude of the net electric field at that central point, greatest first.
Ring charge This was left as an exercise. Consider the following ring of charge with radius R and total charge Q. Show that the expression (or the integral) for the electric field at point P is given by E₂ = kQz (2²+R²)3/2 Disc charge Now let's see how we can use the above result to find the electric field due to a charged disc. Typically for a disc you would need to evaluate a 2D integral, but if you image the disc as an accumulation of ring charges, the integral is effectively 1D¹. R Garea charge density dR ¹Hint: Think about how you would express dQ = odA can be expressed in terms of dR, i.e. how would A and R relate (hence dA with dR).
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Part B:How much charge is on a segment ds? Express your answer in terms of some, all, or none of the variables Q, a, θ, dθ, and the constant π. dQ = ?
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For part (C), x2 = 0 instead of x2 = d
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Problem An electric charge Q is distributed uniformly along a thick and enormously long conducting wire with radius R and length L. Using Gauss's law, what is the electric field at distancer perpendicular to the wire? (Consider the cases inside and outside the wire) Solution To find the electric field inside atr distance from the wire we will use the Gauss's law which is expressed as We will choose a symmetric Gaussian surface, which is the surface a cylinder excluding its ends, then evaluate the dot product to obtain A = (Equation 1) Case 1: Inside the wire Since, r falls inside the wire, then all the enclosed charge must be: denc = On the other hand, the Gaussian surface inside the wire is given by A = Using Equation 1, the electric field in simplified form is E = Case 2: Outside the wire Since, r falls outside the wire, then, all the charge must be enclosed, thus denc = On the other hand, the Gaussian surface outside the wire is given by A = Using Equation 1, the electric field in…