Part 1: A resistor of known resistance R is placed within a circular loop of wire of known radius, a, such that the loop is closed. This loop is placed within a sinusoidal magnetic field of a known amplitude B and period T directed along the loop’s axis of symmetry. Calculate the induced emf across the resistor. Part 2: Now, the situation is the same, but the radius of the loop is also increasing at constant rate x. Calculate the total induced emf across the resistor.

Part 1 - ε = (2π²r²B/T)cosωt

Part 2: ε = B√[(2πr{(πr/T)² + x²}]Sin(ωt + Ф) where Ф = tan⁻¹(πr/xT)

Explanation:

Part 1. Since the magnetic field is sinusoidal and has a period of T and amplitude of B, it is of the form B' = Bsinωt where ω = 2π/T

Now, the induced emf in the circular loop is ε = dΦ/dt where Φ = magnetic flux through circular loop of wire. Φ  = AB' where A = area of loop of wire, A = πr² where r = radius of loop of wire.

So,  ε = dΦ/dt

ε = dAB'/dt

= AdB'/dt

= AdBsinωt/dt

= ωABcosωt

=  (2πAB/T)cosωt    

with A = πr²,

ε = (2π²r²B/T)cosωt

Part 2

If the radius of the loop is increasing at a constant rate x, then the induced emf is

ε = dΦ/dt

= dAB'/dt

= AdB'/dt + B'dA/dt

= πr²dBsinωt/dt  + (Bsinωt)dπr²/dt

= (ωπr²)Bcosωt + (Bsinωt)2πrdr/dt

= (ωπr²)Bcosωt + (Bsinωt)2πrx

= (2π²r²B/T)cosωt + (2πrxB)sinωt

Writing this in compound angle form,

ASin(ωt + Ф) = AsinФcosωt + AcosФsinωt

comparing both expressions,

AsinФ = 2π²r²B/T and AcosФ = 2πrxB

(AsinФ)² + (AcosФ)²  = A²

(2π²r²B/T)² + (2πrxB)²  = A²

B²[(2πr{(πr/T)² + x²}]  = A²

B√[(2πr{(πr/T)² + x²}]  = A

tanФ = AsinФ/AcosФ

tanФ = 2π²r²B/T ÷ 2πrxB

tanФ = πr/xT

Ф = tan⁻¹(πr/xT)

ε = B√[(2πr{(πr/T)² + x²}]Sin(ωt + Ф) where Ф = tan⁻¹(πr/xT)

answer;

d. are made mostly of liquid

explanation;

-the general   composition and structure planets includes; they have a thick gas and liquid layer covering a small,solid core. mostly the gases are hydrogen and helium atmosphere and layers of liquid hydrogen. for instance neptune and uranus are examples of the outer planets, they have a core covered by ice and ammonia.

-neptune's atmosphere is hydrogen and helium with a bit of methane so there is no oxygen to breath and the interior is made up of frozen water and ammonia with a rock and iron core. thus, there is no liquid water or solid surface to survive.

molecules are free to separate from each other and will take on the shape of the container. molecules move completely at random and high speeds. they vibrate in a fixed position and do not move around much. molecules expand to the edge of their container.