The sphere is nonconducting, has a mass of 72.0 g, and a radius 19.0 cm. A flat, compact coil of wire with five turns is wrapped tightly around it, with each turn concentric with the sphere. The sphere is placed on the incline so that the coil is parallel to the plane. You establish a uniform magnetic field of 0.315 T vertically upward in the region of the sphere.

a. What current in the coil will enable the sphere to rest in equilibrium on the inclined plane?
b. Show that the result does not depend on the value of θ.

a) 0.751 A

b) From the derivation below, it is glaring that the current (I) is independent of θ

Explanation:

Given data :

mass of the sphere (m) = 72.0 g = 0.072 kg

number of turns of the wire (N) = 5

radius of the sphere (r) = 19.0 cm = 0.19 m

uniform magnetic field (B) = 0.315

From the given question, the sphere is in translational equilibrium; therefore

 ----------Equation (1)

At the center of the sphere, Lets look at the consideration of the torque; two things is happening;

1) the magnetic field produces a clockwise torque of magnitude μBsinθ

2) the frictional force produces a counterclockwise torque of magnitude

∴

- μBsinθ = 0 ---------- Equation (2)

Substituting from equation (1) to equation (2); we have:

mgsinθr - μBsinθ = 0

Dividing both sides by sinθ; we have:

mgr - μB = 0

mgr = μB

μB = mgr -------------------  Equation (3)

Given that:

μ = NIπr²

Substituting μ = NIπr² into Equation (3); we have:

NIπr²B = mgr

Dividing both sides by r; we have:

NIπrB = mg

Making (I) =

Substituting our given data into equation above; we have:

(I) =

(I) = 0.751 A in counterclockwise direction.

b)

Show that the result does not depend on the value of θ.

From equation (2); we have:

- μBsinθ = 0

We also know that;

;

so replacing    into above equation (2); we have:

-  μBsinθ = 0

= μBsinθ

Dividing both sides by sinθ; we have:

Mgr = μB

However;  μ = NIπr²

Substituting it into the above derivation; we have:

Mgr =  NIπr²B

Dividing both sides by r; we have:

Mg = NIπrB

Making (I) the subject of the formula; we have:

I =

∴ we can conclude that the current is independent on the value of θ.

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: