Magnetic Field inside a Solenoid (Please explain this answer to me)

[Strawberry]

Homework Statement

Q5. A solenoid is 3.0 cm long and has a radius of 0.50 cm. It is wrapped with 500 turns of wire carrying a current of 2.0 A. T

Homework Equations

Magnetic field of an ideal solenoid = permeability constant (1.257E-6) * current * number of turns

The Attempt at a Solution

The given answer is: 4.2 x 10-2 T

I don't understand how to get this answer. From what my book says, the magnetic field is independent of cross-sectional area and only deals with semi-infinite solenoids. Please help!

 

[Strawberry]

err nvm, I figured it out. I was seriously staring at this thing for two hours and then finally figure it out after I post it ><

 

[Moetasim]

First, it is important to understand what a solenoid is. A solenoid is a cylindrical coil of wire that creates a magnetic field when an electric current is passed through it. The magnetic field inside the solenoid is uniform and parallel to the axis of the solenoid.

To calculate the magnetic field inside a solenoid, we can use the equation B = μ0 * n * I, where B is the magnetic field, μ0 is the permeability constant (a physical constant that describes the magnetic properties of a material), n is the number of turns of wire per unit length, and I is the current.

In this case, we are given the length of the solenoid (3.0 cm), the radius (0.50 cm), the number of turns (500), and the current (2.0 A). To find the magnetic field, we need to first calculate the number of turns per unit length, which is given by n = N/L, where N is the number of turns and L is the length of the solenoid. Plugging in the values, we get n = 500/3.0 = 166.67 turns/cm.

Next, we can plug all the values into the equation B = μ0 * n * I. The permeability constant μ0 is a known value (1.257E-6), so we can simply multiply it by the number of turns per unit length (166.67 turns/cm) and the current (2.0 A). This gives us B = (1.257E-6)(166.67)(2.0) = 4.2 x 10-2 T.

Therefore, the magnetic field inside the solenoid is 4.2 x 10-2 T, as given in the answer. This calculation assumes an ideal solenoid, meaning that the length is much greater than the radius and the magnetic field is uniform inside the solenoid. In reality, the magnetic field may vary slightly due to the finite length and radius of the solenoid, but for practical purposes, this calculation is accurate.