# Calculating Magnetic Fields in Solenoids vs. Current Loops and Coils

In summary: Thank you.In summary, there are two different principles at play when using Ampere's Law and Biot-Savart law to find the number of turns in a solenoid. While Ampere's Law can be used for any closed loop, Biot-Savart law is only applicable to individual current elements and must consider the contributions of each along the length of the solenoid. This is why the two solutions for finding the number of turns are different.

## Homework Statement

There is a solenoid, with field B at its center, radius r, length L and maximum current I. Calculate the minimum number of turns the solenoid must have.

## Homework Equations

Ampere's Law: (closed)∫B$\cdot$dl=$\mu$I
n=N/L
Biot-Savart: B=$\mu$IN/(2$\pi$r

## The Attempt at a Solution

Solving Ampere's Law gives you BL=$\mu$IN and so N=BL/($\mu$I).

My question is why can't this same solution be found by solving with Biot-Savart, which is used for coils, according to my book? In that case, my answer is N=2Br/($\mu$I), and thus is off by a factor of 2r/L, or the diameter over the length of the solenoid.

Can someone please explain this to me? Thank you!

Last edited:

Hello,

Thank you for your question. The reason why the solution using Biot-Savart is different from the solution using Ampere's Law is because they are based on different principles.

Ampere's Law is based on the concept of a closed loop and relates the magnetic field around a closed loop to the current passing through the loop. This law is applicable to any closed loop, including a solenoid.

On the other hand, Biot-Savart law is based on the principle that the magnetic field at a point is directly proportional to the current at that point. This law is applicable to a single current carrying wire or a point charge, and is not directly applicable to a solenoid.

When using Biot-Savart law to find the magnetic field inside a solenoid, we must consider the contributions of each individual current element along the length of the solenoid. This is why the solution using Biot-Savart law is different from the solution using Ampere's Law.

I hope this explanation helps to clarify the difference between the two solutions. Please let me know if you have any further questions.

## 1. How do you calculate the magnetic field in a solenoid?

To calculate the magnetic field in a solenoid, you can use the formula B = μnI, where B is the magnetic field, μ is the permeability constant, n is the number of turns per unit length, and I is the current passing through the solenoid. Alternatively, you can use the formula B = μ₀nI, where μ₀ is the permeability of free space.

## 2. What is the difference between a solenoid and a current loop?

A solenoid is a long, cylindrical coil of wire, while a current loop is a circular loop of wire. The main difference between the two is their shape and the direction of the magnetic field they produce. A solenoid produces a uniform magnetic field along its axis, while a current loop produces a magnetic field that is strongest at its center and decreases as you move away from the loop.

## 3. How does the number of turns affect the magnetic field in a solenoid or coil?

The number of turns in a solenoid or coil affects the strength of the magnetic field. As the number of turns increases, the magnetic field also increases. This is because each turn of wire adds to the overall magnetic field, resulting in a stronger field.

## 4. Can you use the same equations to calculate the magnetic field in both solenoids and coils?

Yes, the same equations can be used to calculate the magnetic field in both solenoids and coils. However, keep in mind that the number of turns per unit length and the shape of the coil will affect the final result.

## 5. How do you determine the direction of the magnetic field in a solenoid or coil?

The direction of the magnetic field in a solenoid or coil can be determined using the right-hand rule. Point your thumb in the direction of the current flow and curl your fingers around the coil or solenoid. The direction your fingers curl represents the direction of the magnetic field. Alternatively, you can use the right-hand grip rule, where you point your thumb in the direction of the current and wrap your fingers around the coil or solenoid in the direction of the magnetic field.

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