A current I=1A flows along a thin-walled cylinder of radius R=4cm. What pressure in Pascals do the walls of the cylinder experience? Assume that the current is uniformly distributed. Bonus thing to think about: Is this pressure directed inwards or outwards?
To determine the pressure experienced by the walls of the cylinder, we can use Ampere's Law.
Ampere's Law relates the magnetic field (B) around a closed loop to the current flowing through the loop and the loop's radius. For a long, straight thin-walled cylinder, the magnetic field inside the cylinder is given by:
B = μ₀ * I / (2πR)
where μ₀ is the permeability of free space, I is the current, and R is the radius of the cylinder.
To calculate the pressure on the walls, we need to consider the magnetic force exerted on a small area element of the cylinder's wall.
The force on a small area element dA due to the magnetic field is given by:
dF = B * dA
The pressure (P) is defined as the force per unit area. To find the pressure, we need to divide the force by the area of the cylinder's wall.
The area of the cylinder's wall can be calculated as follows:
A = 2πRH
where H is the height of the cylinder (assuming the whole cylinder is uniformly distributed with the current).
Now, let's put everything together to find the pressure:
1. Calculate the magnetic field B:
B = (μ₀ * I) / (2πR)
2. Calculate the area A:
A = 2πRH
3. Calculate the force on the area element:
dF = B * dA
4. Calculate the pressure P:
P = dF / A
Substitute the values of I = 1A, R = 4cm, H = height, μ₀ = 4π * 10^⁻7 T m/A, and the appropriate units conversions to solve for the pressure in Pascals (Pa).
Regarding the direction of the pressure, it depends on the sign convention used. In this case, since the current flows along the cylinder, the pressure on the walls will be directed inward.