Find the magnetic flux through the area of ​​the rectangular frame

How to find the magnetic flux through a rectangular frame with sides a = 4 cm and b = 8 cm, in which a magnetic field is created by an infinite straight conductor with a current of 6 A, parallel to side b? The conductor and the frame lie in the same plane, and the distance between the conductor and the nearest side of the frame is 2 cm.

The solution to this problem can begin by calculating the magnetic induction B created by the conductor at the point where it is located relative to the frame. To do this, we use the formula for calculating the magnetic induction on the axis of an infinite conductor:

B = (μ₀ * I) / (2 * π * r)

where μ₀ is the magnetic constant (μ₀ = 4 * π * 10^-7 Wb/A*m) I is the current strength in the conductor (I = 6 A) r is the distance from the conductor to the point at which the magnetic induction is calculated.

Since the conductor is parallel to side b of the frame, the magnetic induction at the center of the frame will be equal to B. The distance from the conductor to the nearest side of the frame is 2 cm, and the distance from the conductor to the center of the frame is 4 cm. Thus, the distance from the center of the frame to the nearest side is 2 + 4 = 6 cm.

Therefore, the magnetic induction in the center of the frame will be equal to:

B = (μ₀ * I) / (2 * π * r) = (4 * π * 10^-7 * 6) / (2 * π * 0.06) = 2 * 10^-4 Тл

To find the magnetic flux Ф, you need to multiply the magnetic induction by the area of ​​the frame:

Ф = B * S = 2 * 10^-4 * 0.04 * 0.08 = 6.4 * 10^-7 Vb

Thus, the magnetic flux through the rectangular frame will be equal to 6.4 * 10^-7 Wb.

Find the magnetic flux through the area of ​​the rectangular frame

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To solve this problem, it is necessary to use the Biot-Savart-Laplace law, which allows one to calculate the magnetic field created by a current at a distance from it.

The following data is known from the problem statement:

• current I = 6 A;
• the distance between the current and the nearest side of the frame is r = 2 cm;
• frame side length a = 4 cm;
• frame side length b = 8 cm.

Let's find the magnetic field created by the current at a distance r from it:

B = (μ₀ / 4π) * (2 * I / r)

where μ₀ is a magnetic constant equal to 4π * 10^-7 Wb/(A*m).

Thus, the magnetic field at a distance r = 2 cm from the conductor is equal to:

B = (4π * 10^-7 Wb/(A*m) / 4π) * (2 * 6 A / 0.02 m) = 3 * 10^-3 T

Next, we find the magnetic flux passing through the rectangular frame using the formula:

Φ = B * S * cos(α)

where S is the area of ​​the frame, α is the angle between the direction of the magnetic field and the normal to the area of ​​the frame.

Considering that the magnetic field is directed perpendicular to the plane of the frame, then cos(α) = 0 and the magnetic flux through the frame is zero.

Thus, the answer to the problem: the magnetic flux through a rectangular frame is zero.

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