The electric field in a region of space can be expressed by (285i^ + 1.13×10^3j^ + 796k^) N/C. If a circle of diameter 15.0 cm is placed in the xz-plane of this region, what will be the electric flux through the circle?
The electric flux through the circle can be calculated using the formula:
Φ = E * A * cosθ
where Φ is the electric flux, E is the electric field, A is the area of the circle, and θ is the angle between the electric field vector and the normal to the surface (in this case, the y-axis).
The area of the circle is:
A = π * (d/2)^2
A = π * (15.0 cm / 2)^2
A = π * (7.5 cm)^2
A = π * 56.25 cm^2
A = 176.7145 cm^2 (converted to m^2)
A = 0.01767145 m^2
Now we need to find the angle θ between the electric field vector and the normal to the surface (y-axis). We can do this by finding the magnitude of the electric field vector in the xz-plane and then using the dot product.
E_xz = sqrt(E_x^2 + E_z^2)
E_xz = sqrt((285 N/C)^2 + (796 N/C)^2)
E_xz = sqrt(81,225 N^2/C^2 + 633,616 N^2/C^2)
E_xz = sqrt(714,841 N^2/C^2)
E_xz = 845 N/C
Now we can find the dot product between the electric field vector and the normal to the surface (y-axis).
E_y = 1.13×10^3 N/C
E_dot_n = E_y * 1
E_dot_n = 1.13×10^3 N/C
Now we can find the angle θ using the formula:
cosθ = E_dot_n / E
cosθ = 1.13×10^3 N/C / 845 N/C
cosθ = 1.336
Since cosθ is greater than 1, this means that the electric field vector is not passing through the circle in the xz-plane. Therefore, the electric flux through the circle is zero.
To find the electric flux through the circle, we need to calculate the dot product between the electric field vector and the normal vector of the circle's surface.
The given electric field vector is (285i^ + 1.13×10^3j^ + 796k^) N/C.
The circle lies in the xz-plane, which means its normal vector is in the y-direction, perpendicular to the plane. Therefore, the normal vector can be expressed as (0i^ + 1j^ + 0k^).
The dot product of these two vectors can be calculated using the formula:
Electric Flux = Electric Field dot Product Normal Vector
(Flux) = (Electric Field) dot (Normal Vector)
= (285i^ + 1.13×10^3j^ + 796k^) dot (0i^ + 1j^ + 0k^)
Since the dot product of the electric field vector and the normal vector is equal to the product of their corresponding components, we can calculate it as:
(Flux) = (285 * 0) + (1.13×10^3 * 1) + (796 * 0)
= 0 + 1.13×10^3 + 0
= 1.13×10^3 N/C
Therefore, the electric flux through the circle will be 1.13×10^3 N/C.
To find the electric flux through the circle, we can use Gauss's Law.
Gauss's Law states that the electric flux through a closed surface is equal to the total enclosed charge divided by the permittivity of free space (ε₀).
In this case, we have been given the electric field in the region of space, but we don't have the total enclosed charge. So, to apply Gauss's Law, we need to make an assumption that there is a point charge at the center of the circle that is generating this electric field.
Assuming that there is a point charge at the center of the circle, we can calculate the total charge enclosed within the circle using the following formula:
Q = ε₀ * E * A
where Q is the total charge enclosed, ε₀ is the permittivity of free space (8.85 x 10^-12 C^2/Nm^2), E is the electric field (285i^ + 1.13×10^3j^ + 796k^) N/C, and A is the area of the circle in the xz-plane.
To find the area of the circle, we need to know the radius. The diameter of the circle is given as 15.0 cm, so the radius would be half that value, i.e., 7.5 cm (or 0.075 m).
The area of a circle is given by the formula:
A = π * r^2
Substituting the values, we have:
A = π * (0.075)^2
Calculating the area, we get:
A ≈ 0.01767 m^2
Now, we can substitute the values of ε₀, E, and A into the formula for calculating the total enclosed charge Q:
Q = (8.85 x 10^-12 C^2/Nm^2) * (285i^ + 1.13×10^3j^ + 796k^) N/C * 0.01767 m^2
Simplifying the expression, we get:
Q = (8.85 x 10^-12 C^2/Nm^2) * (0.01767 m^2) * (285i^ + 1.13×10^3j^ + 796k^) N/C
Now, we can calculate the electric flux through the circle using Gauss's Law:
Electric Flux = Q / ε₀
Substituting the calculated value of Q and ε₀, we get:
Electric Flux = [(8.85 x 10^-12 C^2/Nm^2) * (0.01767 m^2) * (285i^ + 1.13×10^3j^ + 796k^) N/C] / (8.85 x 10^-12 C^2/Nm^2)
Simplifying the expression, we find the electric flux through the circle.