The electric flux through each of the six sides of a rectangular box are as follows:
phi 1= +150.0 N*m^2/C; phi 2= +250.0 N*m^2/C;
phi 3= -350.0 N*m^2/C; phi 4= +175.0 N*m^2/C;
phi 5= -100.0 N*m^2/C; phi 6= +450.0 N*m^2/C
How much charge is in this box?
Add them up and apply Gauss' law to the sum.
To find the total charge in the box, we need to sum up the electric flux through each side.
Since electric flux (Φ) is defined as the product of electric field (E) and the area (A) perpendicular to the field, we can calculate the charge (Q) using the formula:
Q = Φ * ε0,
where ε0 is the permittivity of free space.
Let's calculate the charge for each side and sum them up to find the total charge:
Side 1: Φ1 = 150.0 N*m^2/C
Side 2: Φ2 = 250.0 N*m^2/C
Side 3: Φ3 = -350.0 N*m^2/C
Side 4: Φ4 = 175.0 N*m^2/C
Side 5: Φ5 = -100.0 N*m^2/C
Side 6: Φ6 = 450.0 N*m^2/C
Using the formula Q = Φ * ε0, where ε0 = 8.854 x 10^-12 C^2/(N*m^2):
Q1 = Φ1 * ε0
Q2 = Φ2 * ε0
Q3 = Φ3 * ε0
Q4 = Φ4 * ε0
Q5 = Φ5 * ε0
Q6 = Φ6 * ε0
Now we can calculate the total charge, Q_total:
Q_total = Q1 + Q2 + Q3 + Q4 + Q5 + Q6
Calculating the charges:
Q1 = 150.0 N*m^2/C * 8.854 x 10^-12 C^2/(N*m^2) = 1.3266 x 10^-9 C
Q2 = 250.0 N*m^2/C * 8.854 x 10^-12 C^2/(N*m^2) = 2.2135 x 10^-9 C
Q3 = -350.0 N*m^2/C * 8.854 x 10^-12 C^2/(N*m^2) = -3.0879 x 10^-9 C
Q4 = 175.0 N*m^2/C * 8.854 x 10^-12 C^2/(N*m^2) = 1.631 x 10^-9 C
Q5 = -100.0 N*m^2/C * 8.854 x 10^-12 C^2/(N*m^2) = -8.854 x 10^-10 C
Q6 = 450.0 N*m^2/C * 8.854 x 10^-12 C^2/(N*m^2) = 3.9843 x 10^-9 C
Now let's find the total charge:
Q_total = 1.3266 x 10^-9 C + 2.2135 x 10^-9 C - 3.0879 x 10^-9 C + 1.631 x 10^-9 C - 8.854 x 10^-10 C + 3.9843 x 10^-9 C
Q_total = 7.2005 x 10^-9 C
Therefore, the total charge in the box is 7.2005 x 10^-9 Coulombs.
To find the total charge inside the box, we need to use Gauss's Law, which states that the total electric flux through any closed surface is equal to the total charge enclosed by that surface divided by the electric constant (ε0).
The electric flux through each side of the rectangular box represents the electric field passing through that side of the box. To calculate the total electric flux, we need to sum up the fluxes of all six sides.
So, the total electric flux (Φ) through all six sides is:
Φ = Φ1 + Φ2 + Φ3 + Φ4 + Φ5 + Φ6
= +150.0 N*m²/C + 250.0 N*m²/C - 350.0 N*m²/C + 175.0 N*m²/C - 100.0 N*m²/C + 450.0 N*m²/C
Calculating the sum:
Φ = 475.0 N*m²/C
Now, we can use Gauss's Law to find the total charge (Q) enclosed by the box. Gauss's Law states that the total electric flux is equal to the total charge enclosed divided by the electric constant (ε0).
Φ = Q / ε0
Rearranging the equation:
Q = Φ * ε0
The value of ε0 is the electric constant, approximately equal to 8.854 x 10^-12 C²/(N*m²). Substituting the value:
Q = 475.0 N*m²/C * (8.854 x 10^-12 C²/(N*m²))
Calculating the product:
Q = 4.2 x 10^-9 C
Therefore, the total charge inside the rectangular box is approximately 4.2 x 10^-9 Coulombs.