Explain why the work needed to move a particle with charge Q through a potential difference ΔV is W = QΔV, whereas the energy stored in a charged capacitor is = ∆. Where does the factor 1 2 come from?

Question:Explain why the work needed to move a particle with charge Q through a potential difference ΔV is W = QΔV, whereas the energy stored in a charged capacitor is U= 1/2Q∆V. Where does the factor 1/2 come from?

When we move a single charge q through a potential difference ∆v, its potential energy changes by qΔV.

Charging a capacitor involves moving a large number of charges from one capacitor plate to another. If ∆V is the final potential difference on the capacitor, and Q is the magnitude of the charge on each plate, the energy stored in the capacitor is

U = 1/2 Q∆V.

The factor of 1/2 is because, on average, the charges were moved through a potential difference of ½∆v

8 meters above the spring

a solenoid is created by wrapping a l = 90 m long wire around a hollow tube of diameter d = 4.5 cm.   the wire diameter is d = 0.9 mm.   the solenoid wire is then connected to a power supply so that a current of i = 9 a flows through the wire.

randomized variablesl = 90 m

d = 4.5 cm

d = 0.9 mm

i = 9 a                                          

write an expression for the number of turns, n, in the solenoid. you do not need to take into account the diameter of the wire in this calculation.

calculate the number of turns, n, in the solenoid.

write an expression for the length of the solenoid (l2) in terms of the diameter of the hollow tube d.

calculate the length of the solenoid (l2)   in meters.

calculate the magnitude of the magnetic field at the center of the solenoid in teslas.

explanation: