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Transcribed Image Text:### Charge Distributions and Work Done by an Electric Field
**Introduction to Charge Distributions:**
Three charge distributions are depicted in the diagram.
1. **Spherical Shell:**
- A thin, insulating spherical shell with a radius of \(3a\).
- It has a charge of \(-2Q\) uniformly distributed over its surface (where \(Q\) is positive, and the units are in Coulombs).
2. **Circular Arc:**
- A circular arc that spans \(90^\circ\).
- The curve has a non-uniform charge density given by \(\lambda_c = \left(\frac{Q}{a}\right) \sin \phi\), where \(a\) is positive, and \(\phi\) is the angle measured from the \(-x\) axis.
3. **Straight Rod:**
- A thin, straight rod with uniform charge density.
- The rod carries a total charge of \(-3Q\) and extends from \(x = -2a\) to \(x = 2a\) at \(y = 2a\).
**Problem Statement:**
a) Calculate the work done by the electric field in moving a charge \(-Q\) from infinity to the origin. Assume the charge starts and ends at rest.
b) Determine whether the work calculated in part (a) is positive, negative, or zero.
**Guidelines for Solutions:**
- Your answers should include the variables \(Q\), \(a\), and relevant constants such as \(\varepsilon_0\) (the permittivity of free space) and/or \(k\) (Coulomb's constant).
**Diagram Overview:**
- The diagram features a coordinate system with axes labeled \(x\) and \(y\).
- The spherical shell is centered at the origin and has its surface extending out to a radius of \(3a\).
- The circular arc spans the lower right quadrant of the shell from the negative \(x\)-axis, highlighted in red.
- The straight rod is shown as a horizontal line segment above the x-axis from \(-2a\) to \(2a\), at a height of \(y = 2a\) (highlighted in red).
This setup is used to explore the concept of electric fields and the work associated with moving charges in various electric field configurations.
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