If a magnetic flux passes through a circular coil when its diameter D, what should be its diameter (in terms of D) so that only half as much flux passes through it in the same field? Assume that the magnetic field is uniform over the area in both cases.
I am totally lost! I Don't know where to start!
the area should be half the original
Area = pi r^2 = pi D^2/4
new area = pi D^2/8 = pi Dnew^2/4
Dnew ^2 = (1/2) D^2
Dnew = .707 D
ok I am confused, because I have the answer in the back of my book it says it is D/square root of 2
No worries! I can help you break down the problem step by step.
First, let's understand the concept of magnetic flux. Magnetic flux is a measure of the total magnetic field that passes through a specific area. It is represented by the symbol Φ and is given by the equation:
Φ = B * A * cos(θ)
where B is the magnetic field strength, A is the area, and θ is the angle between the magnetic field and the surface normal (which is 90 degrees for a circular coil facing the field).
In this problem, the magnetic field strength B and the area A remain constant. We need to find the relationship between the diameters D1 and D2 such that half as much flux passes through the coil with diameter D2 compared to the coil with diameter D1.
Let's assume that the initial diameter is D1 and the corresponding flux passing through it is Φ1. Then, we can write:
Φ1 = B * A1 * cos(θ)
Now, we need to find the new diameter D2 such that the new flux passing through it, Φ2, is half of Φ1.
Φ2 = (1/2) * Φ1 = (1/2) * B * A1 * cos(θ)
Since the area of a circle is proportional to the square of its diameter, we can express the relationship between the areas as:
A2 = (D2^2/D1^2) * A1
We need to substitute this relationship into the equation for Φ2:
Φ2 = (1/2) * B * [(D2^2/D1^2) * A1] * cos(θ)
Now, we can solve for the relationship between D1 and D2:
(1/2) * B * [(D2^2/D1^2) * A1] * cos(θ) = B * A1 * cos(θ)
Simplifying the equation:
(1/2) * (D2^2/D1^2) = 1
(D2^2/D1^2) = 2
Taking the square root of both sides:
D2/D1 = √2
Therefore, the relationship between the diameters D1 and D2 is:
D2 = D1 * √2
This means that the diameter D2 should be multiplied by the square root of 2 (approximately 1.414) to ensure that only half as much flux passes through the coil in the same magnetic field.
I hope this explanation helps you understand how to approach and solve the problem!