Using gauss's law,calculate the electricfield at a distance r=20cm from a point charge q=4microcoulomb.
(Sphere area @ r) * (E field @ r)
= 4 pi k q
That is Gauss' Law in this situation, and reduces to:
4 pi r^2 * E(r) = 4 pi k q
E(r) = k q /r^2
which you might recognize as Coulomb's Law.
Plug in k and q values to get the E field in Volts/meter (or Newtons/Coulomb). Make sure r in in meters and q is in Coulombs. Look up the value of k if necessary.
To calculate the electric field at a distance r from a point charge q using Gauss's law, you need to follow the steps below:
Step 1: Understand Gauss's Law
Gauss's law relates the electric flux through a closed surface to the charge enclosed within that surface. It can be mathematically expressed as:
∮E · dA = (1/ε₀) * Qenclosed
where ∮E · dA is the electric flux through a closed surface, ε₀ is the permittivity of free space (8.854 × 10⁻¹² C²/(N·m²)), and Qenclosed is the charge enclosed within the surface.
Step 2: Choose a Gaussian Surface
To apply Gauss's law, you need to choose an appropriate Gaussian surface. For a point charge, the most convenient choice is a spherical surface centered around the charge.
Step 3: Determine the Charge Enclosed
The charge enclosed within the chosen Gaussian surface is equal to the total charge, q, as it is a single point charge.
Qenclosed = q
Step 4: Calculate the Electric Flux
To calculate the electric flux through the Gaussian surface, you need to evaluate the dot product of the electric field E and the differential area dA at each point on the surface. However, since the electric field is radially symmetric for a point charge, the dot product simplifies to:
∮E · dA = E * ∮dA
The integral of the differential area, ∮dA, over the entire spherical surface is equal to the surface area of a sphere with radius r:
∮dA = 4πr²
Therefore, the electric flux becomes:
∮E · dA = E * 4πr²
Step 5: Apply Gauss's Law
By applying Gauss's law, you can equate the electric flux to (1/ε₀) times the charge enclosed:
E * 4πr² = (1/ε₀) * Qenclosed
E * 4πr² = (1/ε₀) * q
Step 6: Solve for the Electric Field
Finally, solve the equation for the electric field, E:
E = (1/(4πε₀)) * (q / r²)
Now, substitute the given values:
q = 4 μC = 4 × 10⁻⁶ C
r = 20 cm = 20 × 10⁻² m
E = (1/(4πε₀)) * (q / r²)
E = (1/(4π * 8.854 × 10⁻¹² C²/(N·m²))) * (4 × 10⁻⁶ C / (0.2 m)²)
Evaluate the expression to find the electric field.