A square ammeter has sides of length l = 3.75 cm. The sides of the ammeter are capable of measuring the magnetic field they are subject to. When the ammeter is clamped around a wire carrying a DC current, as shown in the figure, the average value of the magnetic field measured in the sides is 2.73 G. What is the current in the wire?
To find the current in the wire, we can make use of Ampere's Law. Ampere's Law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop.
In this case, the ammeter is in the shape of a square with sides of length l = 3.75 cm. Therefore, the loop formed by the sides of the ammeter can be represented as a square loop.
According to Ampere's Law, the magnetic field measured along the sides of the ammeter is equal to the product of the current passing through the wire and a constant (μ₀/2π) called the permeability of free space.
So we can write the equation as follows:
B (average) = (μ₀/2π) * I / l
where B (average) is the average magnetic field measured, I is the current in the wire, l is the length of each side of the ammeter, and μ₀ is the permeability of free space (constant).
Given that the average magnetic field measured (B average) is 2.73 G (Gauss) and l is 3.75 cm, we plug these values into the equation and solve for I:
2.73 G = (μ₀/2π) * I / 3.75 cm
To solve for I, we need to rearrange the equation:
I = (2.73 G * 3.75 cm * 2π) / μ₀
The value of μ₀ is a constant and is approximately 4π × 10^(-7) T⋅m/A.
Substituting this value:
I = (2.73 G * 3.75 cm * 2π) / (4π × 10^(-7) T⋅m/A)
Now, let's convert the given values to SI units:
2.73 G = 2.73 × 10^(-4) T (1 T = 10^4 G)
3.75 cm = 0.0375 m
Substituting these values:
I = (2.73 × 10^(-4) T * 0.0375 m * 2π) / (4π × 10^(-7) T⋅m/A)
Simplifying the equation:
I = (2.73 × 10^(-4) T * 0.0375 m * 2π) / (4π × 10^(-7) T⋅m/A)
I = 2.73 × 0.0375 / (4 × 10^(-7) A)
I = 0.102375 / (4 × 10^(-7) A)
I ≈ 255.9375 A
Therefore, the current in the wire is approximately 255.9375 A.