Question

A small current element carrying a current of I = 5.00 A is placed at the origin given by d → l = 4.00 m m ^ j Find the magnetic field, d → B , at the locations specified. Enter the correct magnitude and select the direction from the list. If the direction is negative, indicate this by entering the magnitude as a negative number. What is the magnitude and direction of d → B on the x ‑axis at x = 2.00 m ?

Answer 1

Answer:

dB = (-5 × 10⁻⁷k) T

Magnitude of dB = (-5 × 10⁻⁷) T; magnitude is actually (5 × 10⁻⁷) T in the mathematical sense of what magnitude is, but this question instructs to include the negative of the direction of dB is negative.

Direction is in the negative z-direction as evident from the sign on dB's vector notation.

Explanation:

From Biot Savart's relation, the magnetic field is given by

dB = (μ₀I/4πr³) (dL × r)

μ₀ = (4π × 10⁻⁷) H/m

I = 5.0 A

r = (2î) m

Magnitude of r = 2

dL = (4j)

(dL × r) is the vector product of both length of current carrying wire vector and the vector position of the point where magnetic field at that point is needed.

(dL × r) = (4j) × (2î)

​|i j k|

|0 4 0|

|2 0 0|

(dL × r) = (0î + 0j - 8k) = (-8k)

(μ₀I/4πr³) = (4π × 10⁻⁷ × 5)/(4π×2³)

(μ₀I/4πr³) = (6.25 × 10⁻⁸)

dB = (μ₀I/4πr³) (dL × r) = (6.25 × 10⁻⁸) × (-8k)

dB = (-5 × 10⁻⁷k) T

Magnitude of dB = (-5 × 10⁻⁷) T; magnitude is actually (5 × 10⁻⁷) T in the mathematical sense of what magnitude is, but this question instructs to include the negative of the direction of dB is negative.

Direction is in the negative z-direction as evident from the sign on dB's vector notation.

Hope this Helps!!!

Answer 2

When the distance is 4 m, the magnetic field strength is 2.5 x 10⁻⁷ T.

When the distance is 2 m, the magnetic field strength is 5 x 10⁻⁷ T.

Magnitude of magnetic field

The magnitude of magnetic field at any distance from a wire is determined by applying Biot-Savart law,

$B = \frac{\mu_o I}{2\pi r}$

Where;

r is the distance from the conductor

When the distance is 4 m, the magnetic field strength is calculated as;

$B = \frac{(4\pi \times 10^{-7}) \times5 }{2\pi \times 4} \\\\B = 2.5 \times 10^{-7} \ T$

When the distance is 2 m, the magnetic field strength is calculated as;

$B = \frac{(4\pi \times 10^{-7} \times 5}{2\pi \times 2} \\\\B = 5 \times 10^{-7} \ T$

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