At what distance on the axis of a current loop is the magneticfield half the strength of the field at the center of the loop? Give your answer as a multiple of R. z=?R

In the case of a loop or loop, the quantity ds is perpendicular to the distance r, therefore the vector product reduces to the algebraic product and the direction of the field is perpendicular to the current loop

suppose that the spiral eta in the yz plane, therefore the axis is in the x axis

B = μ₀ I/4π ∫ ds / (R² + x²)

The total magnetic field has two components, one parallel to the x axis and another perpendicular, this component is annual when integrating the entire loop, so the total field is

B = Bₓ i^

using trigonometry

Bₓ = B cos θ

we substitute

Bₓ = μ₀ I/4π ∫ ds cos θ / (x² + R²)

the cosine function is

cos θ = R /√(x² + R²)

The differential is

ds = R dθ

we substitute

Bₓ = μ₀ I/4π ∫ (R dθ) R /√( (x² + R²)³ )

we integrate from 0 to 2π

Bₓ =μ₀ I/4π R² / √(x² + R²)³ 2pi

therefore the final expression is

B = μ₀ I R²/ 2√(x² + R²)³ i^

In our case the distance is requested where B is half of B in the center of the bone loop x = 0

Spire center field x=0

B₀ = μ₀ I/2R

Field at the desired point (x)

B = B₀ / 2

we substitute

R² /√(x² + R²)³ = ½ 1 /R

2R³ =√(x² + R²)³

(x² + R²)³ = 4 (R²)³

(x²/R² + 1)³ = 4

The exact result is the solution of this equation, but it is quite laborious, we can find an approximate result assuming that the distance x is much greater than R (x »R)

B = μ₀ I/2x³

we substitute

R² / x³ = 1/2 1 / R

2R³ = x³

x = ∛2 R

x = 1.2599 R

ajeigbeibraheem ajeigbeibraheem · Answer 2 · 2021-11-19T07:53:06+01:00

The distance at which the magnetic field strength is half the strength of the field at the center of the loop in terms of R is 0.766 R

Suppose we consider a magnetic field located at point z on the axis of the current loop with radius R carrying a current (I), then the magnetic field can be represented as:

[tex]\mathbf{B = \dfrac{\mu_o}{2} \dfrac{IR^2}{(z^2+R^2)^{^{\dfrac{3}{2}}}}}[/tex]

And the field situated at the center of the loop is:

[tex]\mathbf{B_{z=0} =\dfrac{\mu_o}{2} \dfrac{I}{R} }[/tex]

Let consider a distance (z) on the axis of the loop, in which the magnetic field as a result of the loop is equal to half the strength of the magnetic field at the center of the loop;

Then;

[tex]\mathbf{B(z) = \dfrac{1}{2}B_{z=0}}[/tex]

[tex]\mathbf{\dfrac{\mu_o}{2} \dfrac{IR^2}{(z^2+R^2)^{\frac{3}{2}}} =\dfrac{1}{2} \Big (\dfrac{\mu_o}{2} \dfrac{I}{R} \Big )}[/tex]

Multiply both sides by (2);

[tex]\mathbf{\dfrac{R^2}{(z^2+R^2)^{\frac{3}{2}}} = \dfrac{1}{2R}}[/tex]

Cross multiply;

[tex]\mathbf{2R^3 = (z^2 +R^2)^{\dfrac{3}{2}}}[/tex]

[tex]\mathbf{4R^6 = (z^2 +R^2)^3}[/tex]

[tex]\mathbf{z = \sqrt{4^{1/3} R^2 -R^2 }}[/tex]

[tex]\mathbf{z = R\sqrt{4^{1/3} -1 }}[/tex]

[tex]\mathbf{z = R\sqrt{0.587401052 }}[/tex]

z = 0.766 R

Therefore, we can conclude that the distance at which the magnetic field strength is half the strength of the field at the center of the loop in terms of R is 0.766 R

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