Question

A magnetic field of 37.2 T has been achieved at the MIT Francis Bitter Magnet Laboratory. Find the current needed to achieve such a field 2.20 cm from a long, straight wire. Express your answer with the appropriate units. Part B Find the current needed to achieve such a field at the center of a circular coil of radius 47.0 cm that has 100 turns. Express your answer with the appropriate units. Find the current needed to achieve such a field near the center of a solenoid with radius 2.20 cm, length 34.0 cm, and 40,000turns. Express your answer with the appropriate units.

Answer 1

To achieve a specific magnetic field at a certain distance from a wire, you can use the formula B = μ0 * (I / 2πr). For a long, straight wire, the current needed can be calculated as I = (B * 2πr) / μ0. For a circular coil, the current needed can be calculated as I = (B * 2R) / (μ0 * N), where R is the radius of the coil. For a solenoid, the current needed can be calculated as I = (B * L) / (μ0 * N), where L is the length of the solenoid.

Step-by-step explanation:

To find the current needed to achieve a specific magnetic field at a given distance from a wire, you can use the formula:

B = μ0 * (I / 2πr)

Where B is the magnetic field, μ0 is the permeability of free space (4π x 10-7 Tm/A), I is the current, and r is the distance from the wire.

For a long, straight wire, the current needed to achieve a magnetic field of 37.2 T at a distance of 2.20 cm can be calculated as:

I = (B * 2πr) / μ0

I = (37.2 T * 2π * 0.022 m) / (4π x 10-7 Tm/A) = 336 A (Amperes)

For a circular coil with a radius of 47.0 cm and 100 turns, you can use the formula:

B = (μ0 * N * I) / (2R)

Where N is the number of turns, I is the current, and R is the radius of the coil.

The current needed to achieve a magnetic field of 37.2 T at the center of the circular coil can be calculated as:

I = (B * 2R) / (μ0 * N)

I = (37.2 T * 2 * 0.47 m) / (4π x 10-7 Tm/A * 100) = 3.71 A (Amperes)

For a solenoid with a radius of 2.20 cm, length of 34.0 cm, and 40,000 turns, you can use the formula:

B = (μ0 * N * I) / (L)

Where N is the number of turns, I is the current, and L is the length of the solenoid.

The current needed to achieve a magnetic field of 37.2 T near the center of the solenoid can be calculated as:

I = (B * L) / (μ0 * N)

I = (37.2 T * 0.34 m) / (4π x 10-7 Tm/A * 40,000) = 1.79 A (Amperes)

Answer 2

For a long, straight wire:
`\(I \approx 1.01 * 10^5 \, \text{A}\)`

For a circular coil:
`\(I \approx 1.58 * 10^4 \, \text{A}\)`

For a solenoid:
`\(I \approx 2.75 \, \text{A}\)`

To find the current needed to achieve a certain magnetic field strength at a certain distance from different configurations of wire setups, we can use Ampère's Law and the formula for the magnetic field created by a straight wire, a circular coil, and a solenoid.

For a long, straight wire carrying current I, the magnetic field at a distance r is given by:

`\[B = (\mu_0 I)/(2\pi r)\]`

Where:

- B is the magnetic field

-
`\(\mu_0\)` is the permeability of free space
`(\(\mu_0 = 4\pi * 10^(-7) \, \text{T}\cdot\text{m/A}\))`

- I is the current

- r is the distance from the wire

Part A: Magnetic field from a long, straight wire at a distance of 2.20 cm

Given:

`\(B = 37.2 \, \text{T}\)`

`\(r = 2.20 \, \text{cm} = 0.022 \, \text{m}\)`

Solving for I:

`\[I = (2\pi rB)/(\mu_0)\]`

`\[I = \frac{2 * \pi * 0.022 \, \text{m} * 37.2 \, \text{T}}{4\pi * 10^(-7) \, \text{T}\cdot\text{m/A}}\]`

`\[I \approx 1.01 * 10^5 \, \text{A}\]`

Part B: Magnetic field at the center of a circular coil with 100 turns and a radius of 47.0 cm

The magnetic field at the center of a circular coil carrying current I with N turns and radius R is given by:

`\[B = (\mu_0 NI)/(2R)\]`

Given:

`\(B = 37.2 \, \text{T}\)`

`\(N = 100\)`

`\(R = 47.0 \, \text{cm} = 0.47 \, \text{m}\)`

Solving for I:

`\[I = (2BR)/(\mu_0 N)\]`

`\[I = \frac{2 * 37.2 \, \text{T} * 0.47 \, \text{m}}{4\pi * 10^(-7) \, \text{T}\cdot\text{m/A} * 100}\]`

`\[I \approx 1.58 * 10^4 \, \text{A}\]`

Part C: Magnetic field near the center of a solenoid with radius 2.20 cm, length 34.0 cm, and 40,000 turns

The magnetic field inside a solenoid carrying current I and having N turns per unit length is given by:

`\[B = \mu_0 nI\]`

Where:

- n is the number of turns per unit length
`(\(n = (N)/(L)\), where \(N\)` is the total number of turns and L is the length of the solenoid)

Given:

`\(B = 37.2 \, \text{T}\)`

`\(R = 2.20 \, \text{cm} = 0.022 \, \text{m}\)`

`\(L = 34.0 \, \text{cm} = 0.34 \, \text{m}\)`

N = 40,000

Calculate n:

`\[n = (N)/(L) = (40,000)/(0.34) \, \text{turns/m}\]`

Solving for I:

`\[I = (B)/(\mu_0 n)\]`

`\[I = \frac{37.2 \, \text{T}}{4\pi * 10^(-7) \, \text{T}\cdot\text{m/A} * (40,000)/(0.34)}\]`

`\[I \approx 2.75 \, \text{A}\]`