A point charge q equals plus 6 n Cis surrounded by an imaginary sphere of radius r equals 40 space c mcentered on the charge as shwon in figure below.
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Calculate the resulting electric flux through the sphere.
To calculate the electric flux through the sphere, we can use Gauss's Law, which states that the electric flux through a closed surface is proportional to the total electric charge enclosed by the surface.
In this case, the imaginary sphere is a closed surface and the charge is enclosed by it. Therefore, the electric flux through the sphere is directly proportional to the charge.
The formula to calculate the electric flux is:
Φ = E * A * cos(θ)
Where Φ is the electric flux, E is the electric field, A is the area of the surface, and θ is the angle between the electric field and the normal vector to the surface.
Since the charge is surrounded by a symmetric sphere, the electric field is uniform at every point on the sphere's surface and is directed radially outward from the charge. Therefore, the angle θ is 0 degrees or 180 degrees, depending on the orientation chosen.
Since we are calculating the electric flux through the entire sphere, we can choose the normal vector to be pointing outward. In this case, θ = 0 degrees, and cos(θ) = 1.
The area of the sphere is given by the formula:
A = 4πr^2
Substituting the radius r = 40 cm into the equation, we have:
A = 4π(40 cm)^2 = 4π(1600 cm^2) = 6400π cm^2
Now we can calculate the electric flux using the formula:
Φ = E * A * cos(θ)
Since the electric field from a point charge is given by Coulomb's law:
E = k * q / r^2
Where k is the Coulomb's constant (k = 8.99 x 10^9 N m^2/C^2), q is the charge (q = 6 nC = 6 x 10^-9 C), and r is the radius (r = 40 cm = 0.4 m).
Substituting these values into the equation, we have:
E = (8.99 x 10^9 N m^2/C^2) * (6 x 10^-9 C) / (0.4 m)^2
E = (8.99 x 10^9 N m^2/C^2) * (6 x 10^-9 C) / (0.16 m^2)
E = (8.99 x 10^9 N m^2) * (6 x 10^-9) / (0.16)
E = 3.3675 x 10^2 N/C
Finally, substituting the values of E, A, and cos(θ) into the equation for electric flux, we have:
Φ = (3.3675 x 10^2 N/C) * (6400π cm^2) * 1
Φ = 2.1556 x 10^6π N m^2/C
Therefore, the resulting electric flux through the sphere is 2.1556 x 10^6π N m^2/C.
To calculate the electric flux through the sphere, we can use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is equal to the electric charge enclosed divided by the permittivity of the medium.
The electric flux, Φ, can be calculated using the formula:
Φ = q_enclosed / ε₀
where q_enclosed is the charge enclosed by the closed surface and ε₀ is the permittivity of free space.
In this case, the charge enclosed is equal to the charge of the point charge, q = +6 nC.
The permittivity of free space, ε₀, is approximately equal to 8.854 x 10^-12 C^2/(N m^2).
Therefore, the electric flux through the sphere is calculated as:
Φ = q / ε₀
= (6 x 10^-9 C) / (8.854 x 10^-12 C^2/(N m^2))
≈ 6.77 x 10^2 N m² / C
So, the resulting electric flux through the sphere is approximately 6.77 x 10^2 N m² / C.
To calculate the resulting electric flux through the sphere, we need to use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by the electric constant.
In this case, we have a point charge q surrounded by an imaginary sphere of radius r. First, we need to find the charge enclosed by the sphere.
The charge enclosed by the sphere is simply the total charge q. In this case, q = +6 nC (nanocoulombs), which is equivalent to 6 x 10^(-9) C.
Next, we need to find the electric constant, which is denoted by ε₀. Its value is 8.854 x 10^(-12) C^2/(Nm^2).
Now we can calculate the electric flux through the sphere using the formula:
Electric flux = (Charge enclosed)/(Electric constant)
Electric flux = (6 x 10^(-9) C) / (8.854 x 10^(-12) C^2/(Nm^2))
Electric flux = (6 x 10^(-9) C) x (1/(8.854 x 10^(-12) C^2/(Nm^2)))
Electric flux ≈ 6.78 x 10^2 Nm^2/C
So, the resulting electric flux through the sphere is approximately 6.78 x 10^2 Nm^2/C.