# A copper ring made of a wire with a diameter of 4 mm is located in a uniform magnetic field, the magnetic

**A copper ring made of a wire with a diameter of 4 mm is located in a uniform magnetic field, the magnetic induction of which changes in absolute value at a speed of 1.48 tesla per second. The plane of the ring is perpendicular to the vector of magnetic induction. What is the diameter of the ring if the induction current is 8 amperes.**

d = 4 mm = 4 * 10 ^ -3 m.

ΔB = 1.48 T.

t = 1 s.

∠β = 90 “.

I = 8 A.

ρ = 0.0175 * 10 ^ -6 Ohm * m.

D -?

Faraday’s law

According to Faraday’s law of electromagnetic induction, when the magnetic flux F piercing the circuit changes, the EMF of induction appears in the circuit.

EMF of induction is directly proportional to the rate of change of the magnetic flux ΔF / t: EMF = – ΔF / t.

The “-” sign indicates the direction of the induction current.

The magnetic flux Ф is determined by the formula: Ф = В * S * cos α, where

B is the magnetic induction of the field;

S is the area of the contour;

∠α is the angle between B and the perpendicular to the plane of the contour.

∠α = 90 “- ∠β = 90” – 90 “= 0”.

Let us express the change in magnetic flux ΔF = Δ (B * S * cos α) = Δ B * S * cos0 “= Δ B * S.

Since the contour has the shape of a ring, the area of the contour S will be determined by the formula: S = P * D ^ 2/4, where D is the contour diameter.

The formula for determining the change in magnetic flux will take the form: ΔF = Δ B * P * D ^ 2/4.

The formula for Faraday’s law will be: EMF = Δ B * P * D ^ 2/4 * t.

Ohm’s law

Ohm’s law is true for a conductor through which an electric current flows.

According to Ohm’s law, the current in the conductor I is directly proportional to the EMF and inversely proportional to the resistance of the conductor R: I = EMF / R.

The resistance of a cylindrical conductor R is determined by the formula: R = ρ * L / s, where

ρ is the resistivity of copper;

L is the length of the conductor;

s is the cross-sectional area of the conductor.

The length of the conductor L is determined by the formula: L = P * D, where D is the diameter of the ring.

The cross-sectional area of the conductor is determined by the formula: s = P * d ^ 2/4, where d is the diameter of the copper wire.

R = ρ * P * D * 4 / P * d ^ 2 = ρ * D * 4 / d ^ 2.

Ohm’s law formula will take the form: I = Δ B * P * D ^ 2 * d ^ 2/4 * t ρ * D * 4 = Δ B * P * D * d ^ 2/16 * t * ρ.

Let us express the diameter of the ring: D = 16 * t * ρ * I / Δ B * P * D * d ^ 2.

D = 16 * 1 s * 0.0175 * 10 ^ -6 Ohm * m * 8 A / 1.48 T * 3.14 * (4 * 10 ^ -3 m) ^ 2 = 0.03 m.

Answer: the diameter of the ring is D = 0.03 m.