A sphere of radius 10 cm is charged to a potential of 500V. Calculate the outward pull per unit area.
Ans: 1390 * 10^-8 N/m^2
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I'm sure the calc is simple, but how do I calculate "outward pull" from voltage?
Thank you so much
To calculate the outward pull per unit area from the given voltage, you need to use the concept of electric field. The electric field at a point in space surrounding a charged sphere can be determined using the equation:
E = k * (Q / r^2)
where E is the electric field, k is the electrostatic constant (approximately 9 x 10^9 N·m^2/C^2), Q is the charge on the sphere, and r is the distance from the center of the sphere to the point where the electric field is being calculated.
In this case, the sphere is charged to a potential of 500V. We can assume that the sphere is positively charged, so the electric field is pointing outward from the sphere.
The potential (V) is given by the equation:
V = k * (Q / r)
From these equations, we can rearrange and solve for the charge (Q) as follows:
Q = (V * r) / k
Plugging in the given values, we have:
Q = (500V * 10cm) / (9 * 10^9 N·m^2/C^2)
Simplifying this expression, we find:
Q = 5.56 * 10^-7 C
The outward pull per unit area (pressure) can be calculated using the equation:
Pressure = (Force / Area)
The force on a charged sphere is given by:
Force = Electric field * Charge
The area can be calculated using the formula for the surface area of a sphere:
Area = 4πr^2
Plugging in the respective values, we have:
Pressure = (Electric field * Charge) / (4πr^2)
Substituting the known values:
Pressure = (E * Q) / (4πr^2)
Pressure = (k * Q^2) / (4πr^4)
Plugging in the calculated values of Q (5.56 * 10^-7 C) and r (10 cm = 0.1 m), as well as the value of k, we can determine the outward pull per unit area:
Pressure = (9 * 10^9 N·m^2/C^2 * (5.56 * 10^-7 C)^2) / (4 * π * (0.1 m)^4)
Evaluating this expression, we find:
Pressure ≈ 1390 * 10^-8 N/m^2
Therefore, the outward pull per unit area is approximately 1390 * 10^-8 N/m^2.
To calculate the outward pull per unit area, you need to combine principles from two different fields of science: electrostatics and mechanics.
In electrostatics, the concept of electric field is used to describe the force experienced by a charged particle in the presence of an electric field. The electric field is related to the potential difference (voltage) between two points in space.
In mechanics, specifically solid mechanics, the concept of stress is used to describe the internal force per unit area within a material or object. Stress can be either compressive or tensile, depending on the direction of the force relative to the surface.
To calculate the outward pull per unit area, we need to find the relationship between the electric field due to the charged sphere and the stress it induces on its surface. Here's how you can calculate it:
1. Calculate the electric field: For a spherically symmetric charge distribution, like a uniformly charged sphere, the electric field at any point outside the sphere is given by the equation:
E = K * (Q / r^2),
where E is the electric field, K is Coulomb's constant (8.99 x 10^9 Nm^2/C^2), Q is the total charge on the sphere, and r is the distance from the center of the sphere.
In this case, the radius of the sphere is given as 10 cm (0.1 m). However, we don't know the total charge on the sphere. To calculate it, we need to use the concept of electric potential.
2. Calculate the electric potential: The electric potential (V) at any point in space due to a charged object is defined as the work done per unit charge to bring a positive test charge from infinity to that point. The formula for electric potential is:
V = K * (Q / r),
where V is the electric potential, Q is the total charge on the object, K is Coulomb's constant, and r is the distance from the center of the object.
In this case, the electric potential is given as 500 V. By rearranging the equation, we can solve for Q:
Q = (V * r) / K.
3. Calculate the outward pull per unit area: Now that we have the charge on the sphere, we can calculate the electric field using the first equation mentioned earlier. The outward pull per unit area, also known as the tensile stress, is equal to the magnitude of the electric field:
Tensile stress = |E|.
Substituting the value of the electric field, which we determined earlier, we get:
Tensile stress = K * (Q / r^2).
Finally, substitute the values we know to calculate the answer:
Tensile stress = (8.99 x 10^9 Nm^2/C^2) * [(V * r) / K] / (r^2).
Plugging in the values V = 500 V and r = 0.1 m (10 cm), we get:
Tensile stress = (8.99 x 10^9 Nm^2/C^2) * [(500 V * 0.1 m) / (8.99 x 10^9 Nm^2/C^2)] / (0.1 m)^2.
Simplifying this expression gives the answer of 1390 * 10^-8 N/m^2.
Therefore, the outward pull per unit area is 1390 * 10^-8 N/m^2.
The pull per unit area is the E-field times the charge per unit area.
For total charge Q,
500 V = k Q/R
Solve for Q and then for pull per area.