Chapter 21: Problem 52
A straight, nonconducting plastic wire 8.50 cm long carries a charge density of \(+\)175 nC\(/\)m distributed uniformly along its length. It is lying on a horizontal tabletop. (a) Find the magnitude and direction of the electric field this wire produces at a point 6.00 cm directly above its midpoint. (b) If the wire is now bent into a circle lying flat on the table, find the magnitude and direction of the electric field it produces at a point 6.00 cm directly above its center.
Short Answer
Step by step solution
Calculate the linear charge density
Determine the electric field at a point above a straight wire
Plug values to find electric field for part (a)
Determine the electric field at a point above a circular wire
Use formula for the electric field of a ring
Calculate the electric field for part (b)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Linear Charge Density
- Units: Usually expressed in terms of charge per unit length, like nanocoulombs per meter \(\text{ nC/m}\).
- Uniform distribution: The charge must be distributed evenly along the wire or circle.
The Role of Permittivity of Free Space
- Significance: It affects the strength of the electric field produced by a charged object.
- In equations: Used in formulas such as \( E = \frac{1}{4\pi\varepsilon_0} \cdots \).
Calculating Electric Field Due to a Wire
- \( L \): Length of the wire.
- \( r \): Distance from the wire.
- \( \lambda \): Charge density.
Electric Field Due to a Ring
The formula for the electric field at a point directly above the center of the ring is:\[E = \frac{1}{4\pi \varepsilon_0} \cdot \frac{Qz}{(z^2 + R^2)^{3/2}}\]
- \( Q \): Total charge on the ring.
- \( z \): Height above the center of the ring.
- \( R \): Radius of the ring, found via \( R = \frac{L}{2\pi}\).
Exploring Symmetry in Electric Fields
- For the straight wire: The electric field is symmetric around the axis perpendicular to the wire, meaning it is radial and depends inversely on distance.
- For the ring: When the wire is bent into a circle, symmetry allows us to focus on the vertical component of the electric field, as horizontal components cancel out.