50.0pJ of energy is stored in a 3.00cm × 3.00cm × 3.00cm region of uniform electric field. What is the strength of the field?
Well, to find the strength of the electric field, we need to divide the energy stored by the volume of the region. But hold on a second, I have a joke for you:
Why don't scientists trust atoms?
Because they make up everything! Ah, the classic chemistry humor.
Now let's get back to your question. The volume of the region is 3.00cm * 3.00cm * 3.00cm, which is 27.00 cm^3. But let's convert that to a more convenient unit like meters to avoid confusion.
So, 27.00 cm^3 is equal to 0.000027 m^3. Now let's divide the energy (50.0 pJ) by the volume (0.000027 m^3) to find the strength of the electric field. Are you ready? Drumroll, please...
The strength of the electric field is 1.85 GJ/m^3. Don't worry if it seems like a lot; electric fields can be quite shocking!
To find the strength of the electric field, we need to use the formula for the electric potential energy:
Potential Energy (PE) = Electric Field Strength (E) * Charge (q) * Distance (d)
In this case, the potential energy (PE) is given as 50.0 pJ (picojoules). We are also given the dimensions of the region of the electric field, which is a cube with side length 3.00 cm.
Since the field is uniform, we can assume that the strength of the field is constant throughout the region. Therefore, we can take any point inside the region to calculate the electric field strength.
Let's choose the center of the cube, which has a distance of half of each side length from any face.
The distance (d) from the center to any face is half of the side length, so d = 1.50 cm.
Now we can rearrange the formula to solve for the electric field strength (E):
E = PE / (q * d)
However, we don't have information about the charge (q) in this problem. We need either the charge or the value of potential energy in order to calculate the field strength accurately.
To find the strength of the electric field in this case, we can use the formula:
E = sqrt(2U / ε0 * A),
where E is the strength of the electric field, U is the stored energy, ε0 is the permittivity of free space (8.85 x 10^-12 C^2/N*m^2), and A is the area of the region.
First, let's convert the energy U from pJ to Joules (J). Recall that 1 pJ = 10^-12 J. Therefore,
U = 50.0pJ = 50.0 x 10^-12 J.
Next, let's calculate the area A. The given region is a cube with sides measuring 3.00 cm. Since the sides are equal, the area of each face is 3.00 cm × 3.00 cm = 9.00 cm^2.
However, we need to convert cm^2 to m^2 to keep the units consistent. Recall that 1 cm^2 = (1/100)^2 m^2 = 10^-4 m^2. Therefore,
A = 9.00 cm^2 = 9.00 x 10^-4 m^2.
Now, let's substitute the values into the formula to find the strength of the electric field:
E = sqrt(2 * U / ε0 * A)
E = sqrt((2 * 50.0 x 10^-12 J) / (8.85 x 10^-12 C^2/N*m^2) * (9.00 x 10^-4 m^2)).
Calculating this expression using a calculator, we find:
E ≈ 1.726 x 10^13 N/C.
Therefore, the strength of the electric field is approximately 1.726 x 10^13 N/C.
energy/volume = 0.5 * epsilon * E^2
E = (50)/(0.03)^3 = 0.5 * *8.85E-12)E^2
50/(0.03)^3 = 4.425 * E^2
E = sqrt (50/(0.03)^3/4.425
E = 646.9138997 V/m