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A cylindrical Gaussian surface of length L and radius R encloses a charge Q unif

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Question: A cylindrical Gaussian surface of length L and radius R encloses a charge Q uniformly distributed along its length. What is the electric field at a distance R from the axis of the cylinder?

Options:

  1. Q/(2πε₀R)
  2. Q/(4πε₀R²)
  3. 0
  4. Q/(ε₀L)

Correct Answer: Q/(2πε₀R)

Solution: 

Using Gauss\'s law, the electric field outside the cylinder is E = Q/(2πε₀R).

A cylindrical Gaussian surface of length L and radius R encloses a charge Q unif

Practice Questions

Q1
A cylindrical Gaussian surface of length L and radius R encloses a charge Q uniformly distributed along its length. What is the electric field at a distance R from the axis of the cylinder?
  1. Q/(2πε₀R)
  2. Q/(4πε₀R²)
  3. 0
  4. Q/(ε₀L)

Questions & Step-by-Step Solutions

A cylindrical Gaussian surface of length L and radius R encloses a charge Q uniformly distributed along its length. What is the electric field at a distance R from the axis of the cylinder?
  • Step 1: Understand that we have a cylindrical Gaussian surface with a charge Q distributed along its length.
  • Step 2: Identify that we want to find the electric field at a distance R from the axis of the cylinder.
  • Step 3: Recall Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (ε₀).
  • Step 4: For a cylindrical surface, the electric field (E) is uniform and points radially outward, so we can express the electric flux as Φ = E × A, where A is the surface area of the cylindrical Gaussian surface.
  • Step 5: The surface area A of the cylindrical Gaussian surface is 2πRL, where R is the radius and L is the length of the cylinder.
  • Step 6: Set up Gauss's law: Φ = Q_enc / ε₀, where Q_enc is the charge enclosed by the Gaussian surface. Since the charge is uniformly distributed, Q_enc = Q.
  • Step 7: Substitute the expression for electric flux into Gauss's law: E × (2πRL) = Q / ε₀.
  • Step 8: Solve for the electric field E: E = Q / (2πε₀R).
  • Gauss's Law – A fundamental principle in electromagnetism that relates the electric field to the charge enclosed by a surface.
  • Cylindrical Symmetry – The property of a system where the physical characteristics are invariant under rotation around a central axis.
  • Electric Field Calculation – The process of determining the electric field produced by a charge distribution using appropriate laws and symmetries.
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