Consider a thin, infinitely long conducting ribbon that carries a uniform current density j (current per unit area). The width of the ribbon is w and its thickness s is extremely small (s≪w). P is a point in the plane of the ribbon, at a large distance (x≫s) from the ribbon edge.
What is the magnitude of the magnetic field B (in T) at point P for the following values of w , j, s and x?
w= 5 cm; s= 0.1 cm; j=1A/m2 and x= 21 cm
honour code breach!!
To find the magnitude of the magnetic field B at point P, we can use the Biot-Savart law. According to the Biot-Savart law, the magnetic field at a point due to a current element is given by:
dB = (μ₀/4π) * (Idl × r)/r²
where dB is the differential magnetic field, μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), I is the current, dl is the length of the current element, r is the distance between the current element and the point where the magnetic field is being measured, and × represents the cross product.
In this case, we have a current density j instead of a specific current I. The current density j is defined as the current per unit area, so we can write:
I = j * A
where A is the area of the ribbon. In this case, the ribbon is infinitely long and has a width w and thickness s. Therefore, the area A is given by:
A = w * s
Now we can rewrite the Biot-Savart law using the current density:
dB = (μ₀/4π) * (j * A * dl × r)/r²
Since the ribbon is thin and infinitely long, we can assume that the magnetic field will be perpendicular to the surface of the ribbon at every point. Therefore, we can simplify the cross product to just the dl component:
dB = (μ₀/4π) * (j * A * dl)/r²
To find the total magnetic field B at point P, we need to integrate the differential magnetic fields over the entire length of the ribbon. Since the ribbon is infinitely long, we integrate from -∞ to +∞. So:
B = ∫(μ₀/4π) * (j * A * dl)/r²
Now let's solve the equation using the given values:
w = 5 cm = 0.05 m
s = 0.1 cm = 0.001 m
j = 1 A/m²
x = 21 cm = 0.21 m
First, we need to find the length of the ribbon, which is infinite. However, since x is very large compared to s, we can approximate the length of the ribbon within the region that contributes to the magnetic field at point P as x. Therefore, we can use x as the upper limit for the integration.
B = ∫(μ₀/4π) * (j * A * dl)/r²
= (μ₀/4π) * (j * A) * ∫dl/r² (from -∞ to x)
= (μ₀/4π) * (j * A) * ∫dx/x² (from -∞ to x)
Integrating ∫dx/x² gives us -1/x evaluated from -∞ to x. Since we are dealing with an infinite length, we take the limit as x approaches infinity:
B = (μ₀/4π) * (j * A) * (-1/x) (from -∞ to x)
= -(μ₀/4π) * (j * A) * (1/x)
Now substitute the values:
B = -(μ₀/4π) * (j * A) * (1/x)
= -(4π × 10⁻⁷ T·m/A) / (4π) * (1 * 0.05 m * 0.001 m) * (1/0.21 m)
= -10⁻⁷ * (0.05 * 0.001) / (0.21)
Finally, we take the absolute value of B since we are interested only in its magnitude:
|B| = |0.05 * 0.001 / 0.21|
Evaluating the expression:
|B| = |0.00005 / 0.21|
Therefore, the magnitude of the magnetic field B at point P is approximately:
|B| ≈ 0.000238 T (or 2.38 × 10⁻⁴ T)