A semi-circular ring of radius R carries a uniform linear charge density ??. The tiny chunk of charge dQ produces the electric field dE at point P in the direction shown in the figure below. a. Write the expression for dEx and dEy in terms of R, ??, ? and constants. b. Write the integrals, including limits, that would allow you to calculate Ex and Ey at point P. It is not necessary to evaluate the integrals. c. Write the integral, including limits, that would allow you to calculate the electric potential at P in terms of R, ??, ? and constants. It is not necessary to evaluate the integral.
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Step 1: The expression for \(dE_x\) and \(dE_y\) in terms of \(R\), \(A\), and constants is given by: \[dE_x = \frac{K \lambda \cos(\theta) d\theta}{R}\] \[dE_y = -\frac{K \lambda \sin(\theta) d\theta}{R}\] Show more…
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Consider a circular ring of radius $r$, uniformly charged with linear charge density $\lambda$. Find the electric potential at a point on the axis at a distance $x$ from the centre of the ring. Using this expression for the potential, find the electric field at this point
Example 3: A semicircular ring of radius R carries a uniform line of charge λ. Find the electric field at the center of the semicircle. 1. a) Make a good drawing. Let the open side of the semicircle face the -x direction. b) Select the point in space where the electric field is to be determined (call it P). c) Select a small element of charge from the distribution. d) Draw the electric field at point P due to your small charge element. 2. From Coulomb’s law, write down the electric field due to the arbitrary charge element. 3. Derive the equation that relates the charge dq to the charge density. 4. Change the integral to integrate over spatial variables (what are the new limits of integration?). Perform the integration to find the total x component of the electric field.
Mary W.
(II) A very long solid nonconducting cylinder of radius $R_{1}$ is uniformly charged with a charge density $\rho_{\mathrm{E}} .$ It is surrounded by a concentric cylindrical tube of inner radius $R_{2}$ and outer radius $R_{3}$ as shown in Fig. $36,$ and it too carries a uniform charge density $\rho_{\mathrm{E}} .$ Determine the electric field as a function of the distance $R$ from the center of the cylinders for $(a) 0<R<R_{1},(b) \quad R_{1}<R<R_{2}$ (c) $R_{2}<R<R_{3},$ and $(d) R>R_{3} .(e)$ If $\rho_{\mathrm{E}}=15 \mu \mathrm{C} / \mathrm{m}^{3}$ and $\quad R_{1}=\frac{1}{2} R_{2}=\frac{1}{3} R_{3}=5.0 \mathrm{cm}$ plot $E$ as a function of $R$ from $R=0$ to $R=20.0 \mathrm{cm} .$ Assume the cylinders are very long compared to $R_{3}$ .
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