What is the charge of each cylinder given the information provided?
Calculating Charge of Each Cylinder:
Each of the three solid plastic cylinders has a radius of 2.73 cm and a length of 6.48 cm. Cylinder (a) has a charge with uniform density of 13.7 nC/m² on its entire surface, cylinder (b) has a charge with uniform density of 13.7 nC/m² only on its curved lateral surface, and cylinder (c) has a charge with uniform density of 490 nC/m³ throughout the plastic.
The surface area of each cylinder can be calculated using the formula: S = 2πrh + 2πr², where h is the height and r is the radius of the cylinder. Substituting the values, we get:
S = 2π(2.73)(6.48) + 2π(2.73)²
S = 106.822 + 59.049
S = 165.871 cm².
The volume of each cylinder can be calculated as V = πr²h. For cylinder (c), the charge Q can be calculated using the formula Q = ρV, where ρ is the charge density and V is the volume of the cylinder:
Q = (490 nC/m³) × π(2.73 cm)²(6.48 cm)
Q = 584.634 nC.
For cylinder (a), the charge Q is uniformly distributed across the surface and can be calculated using the formula Q = σS, where σ is the surface charge density and S is the surface area of the cylinder:
Q = (13.7 nC/m²) × (165.871 cm² / (100 cm/m)²)
Q = 22.747 nC.
For cylinder (b), the charge Q is only on its curved lateral surface. The surface area to be used in the formula for cylinder (b) is the curved lateral surface area:
Q = (13.7 nC/m²) × (106.822 cm² / (100 cm/m)²)
Q = 14.655 nC.
Therefore, the charge of each cylinder is 584.634 nC for cylinder (c), 22.747 nC for cylinder (a), and 14.655 nC for cylinder (b).