How to Calculate Electric Field from Rods?

In summary, the problem involves two long, thin parallel rods joined by a semicircular piece of radius b and a charge of uniform linear density λ deposited along the whole filament. We are asked to show that the field E of this charge distribution vanishes at the point C by comparing the contribution of the element at A to that of the element at B. The analogous two-dimensional setup involves a cylinder and a hemispherical end cap with uniform surface charge density σ. Using the result from part (a), we can determine that the field at the analogous point C is directed upward, downward, or is zero without any calculations needed. The distance from the dashed line and the length of the element in that direction are important factors in determining the charge
  • #1
FS98
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Homework Statement



(a) Two long, thin parallel rods, a distance 2b apart, are joined by a semicircular piece of radius b, as shown in Fig. 1.44. Charge of uniform linear density λ is deposited along the whole filament. Show that the field E of this charge distribution vanishes at the point C. Do this by comparing the contribution of the element at A to that of the element at B which is defined by the same values of θ and dθ.
(b) Consider the analogous two-dimensional setup involving a cylinder and a hemispherical end cap, with uniform surface charge density σ. Using the result from part (a), do you think that the field at the analogous point C is directed upward, downward, or is zero? (No calculations needed!)

Homework Equations



F = kq/d^2

The Attempt at a Solution



I have no idea how to begin this problem. Can somebody explain to me where I can start? I imagine I have to use the equation above and find that the values from each part of the rods cancel out, but I don’t know how without knowing the charge in either case or the distance in the one case.
 

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  • #2
How long is the shaded element at A? So how much charge on it?
How far is it from C? What is the strength of field it exerts there?

Then the same questions for the other shaded element. For this, it might help to consider the distance of the element from the dashed line, and how that distances changes as the angle increases by dθ.
 
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  • #3
haruspex said:
How long is the shaded element at A? So how much charge on it?
How far is it from C? What is the strength of field it exerts there?

Then the same questions for the other shaded element. For this, it might help to consider the distance of the element from the dashed line, and how that distances changes as the angle increases by dθ.

I believe the distance to A is b and the distance to B is b/(cos(θ))

I understand that the charge on each part of the rod is related to the length of that part and that there should be a greater charge at B because of the longer distance. I’m not sure how to calculate these charges though.

Edit: I got a charge of λbdθ for B and λbdθ/cos(θ) for A by multiplying the small angle by the distance. When I try to calculate the electric field at C from these two points, they don’t cancel out. The values I got are kλdθ/b and kcos(θ)λdθ/b. Do you know what I did wrong?
 
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  • #4
FS98 said:
distance to B is b/(cos(θ))
No, I asked for the distance from the dashed line. That is a distance along the straight arm of the wire.
You need the length of the element in that direction. This will be the change in the distance along the arm for a small change dθ in the angle.
 

Related to How to Calculate Electric Field from Rods?

1. What is an electric field from rods?

An electric field from rods refers to the distribution of electric charges along a rod or cylinder, which creates an electric field in the surrounding space. This field can be either attractive or repulsive, depending on the type of charge on the rod.

2. How is the strength of an electric field from rods measured?

The strength of an electric field from rods is measured in units of volts per meter (V/m). This value represents the force per unit charge experienced by a test charge placed in the field.

3. What factors affect the strength of an electric field from rods?

The strength of an electric field from rods is affected by the distance from the rod, the amount of charge on the rod, and the orientation of the rod relative to the test charge. Additionally, the properties of the surrounding medium, such as its dielectric constant, can also impact the strength of the field.

4. How is the direction of an electric field from rods determined?

The direction of an electric field from rods is determined by the direction in which a positive test charge would be pushed or pulled if placed in the field. The direction is always perpendicular to the equipotential lines, which are imaginary lines that connect points with equal electric potential.

5. What are some real-world applications of electric fields from rods?

Electric fields from rods have many practical applications, such as in electrostatic precipitators for air pollution control, Van de Graaff generators for creating high voltage, and in electrostatic spraying for coating surfaces. They are also essential in understanding the behavior of charged particles in devices like cathode ray tubes and particle accelerators.

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