Consider the uniform electric field
E = (8.0ĵ + 4.0 k) ✕ 103 N/C.
What is its electric flux (in N · m2/C) through a circular area of radius 8.0 m that lies in the xy-plane? (Enter the magnitude in N · m2/C.)
E = (8.0ĵ + 4.0 k) * 10^3 N/C I think you mean.
The j component is IN the plane. It does NOT go THROUGH.
Only the k component matters, and it is exactly perpendicular to the plane so it is exactly what you are looking for.
4.0 * 10^3 N/C Times the area in the plane which is pi R^2
4.0 * 10^3 Newtons /Coulomb * pi * 8^2 meters^2
= about 804*10^3 N m^2/C
= 8.04*10^5 N m^2/C
Well, to calculate the electric flux, we can use the formula:
Φ = E * A * cos(θ)
Where:
Φ is the electric flux,
E is the electric field,
A is the area, and
θ is the angle between the electric field and the area vector.
In this case, the electric field is given by E = (8.0ĵ + 4.0k) * 10^3 N/C.
Now, the circular area lies flat in the xy-plane, so the electric field is perpendicular (θ = 90°) to the area vector. Therefore, cos(90°) = 0.
Since cos(θ) = 0, the flux Φ becomes:
Φ = E * A * 0 = 0.
So, the magnitude of the electric flux through the circular area is 0 N · m^2/C.
I guess you could say the electric flux through this area is a big fat zero. Maybe there's an electric ghost hiding in there, who knows!
To find the electric flux through a circular area, we can use the formula:
Electric Flux = Electric Field * Area * cos(theta)
In this case, the electric field is given as E = (8.0ĵ + 4.0k) ✕ 10^3 N/C.
The area of the circular area is given as the radius, which is 8.0 m.
To find the magnitude of the electric field, we can take the dot product of the electric field and the unit normal vector of the circular area, which is in the direction of the z-axis (k direction).
The dot product of the electric field and the normal vector is:
E • A = (8.0ĵ + 4.0 k) ✕ 10^3 N/C • 8.0 m • cos(θ)
Since the circular area lies in the xy-plane, the angle between the electric field and the normal vector (θ) is 90 degrees.
Therefore, cos(θ) = cos(90°) = 0.
Multiplying the dot product by 0 will result in the electric flux being 0 N · m²/C.
To find the electric flux through a circular area, we need to use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is equal to the total charge enclosed divided by the permittivity of free space.
In this case, we are given a uniform electric field E and a circular area. The electric field is given as E = (8.0ĵ + 4.0 k) ✕ 103 N/C.
To find the electric flux, we first need to find the total charge enclosed by the circular area. Since we are not given the charge distribution, we can assume that the charge is uniformly distributed over the circular area. To find the charge, we can use the formula:
Q = σ × A,
where Q is the charge, σ is the charge density (assumed to be uniform), and A is the area of the circular region.
The area of a circle is given by the formula:
A = π × r^2,
where r is the radius.
In this case, the radius is given as 8.0 m, so the area A is:
A = π × (8.0 m)^2.
Once we have the area A, we can find the total charge Q.
Next, we can calculate the electric flux using Gauss's Law:
Φ = (Q / ε₀),
where Φ is the electric flux, Q is the total charge enclosed by the circular area, and ε₀ is the permittivity of free space.
Finally, we can substitute the known values into the equation and calculate the electric flux in N · m^2/C.