Question

Awire of length 20.0 cm lies along the x-axis with the center of the wire at the origin. The wire carries current I=8.00 A in the - - -direction. What is the magnitude B of the magnetic field of the wire at the point y=5.00 cm on the y-axis? Express your answer with the appropriate units. Part B What is the percent difference between the answer in A and the value you obtain if you assume the wire is infinitely long and use equation: B= Ito calculate B? Express your answer in percents.

Answer 1

To find the magnitude of the magnetic field B of the wire at a specific point, we can use the Biot-Savart Law. In this case, the wire is along the x-axis with current in the - - -direction. The magnetic field can be calculated using the formula B = μ₀(I/(2πr)). The percent difference between the calculated magnetic field and the value assuming the wire is infinitely long can also be found.

Step-by-step explanation:

To find the magnitude of the magnetic field B of the wire at the point y=5.00 cm on the y-axis, we can use the Biot-Savart Law. The formula for the magnetic field of a long straight wire is given by B = μ₀(I/(2πr)), where μ₀ is the permeability of free space, I is the current, and r is the distance from the wire. In this case, the wire is laying along the x-axis with the current flowing in the - - -direction, so the magnetic field will have a y-component. Plugging in the values, we have B = (4π×10^-7 T·m/A)(8.00 A/(2π(0.2 m))), which simplifies to B = 0.01 T.

To calculate the percent difference between the answer in part A and the value obtained by assuming the wire is infinitely long and using the equation B = I, we can use the formula for percent difference. The formula for percent difference is |(value1 - value2)/value1| x 100%. In this case, value1 is the magnitude of the magnetic field calculated in part A (0.01 T), and value2 is the magnitude of the magnetic field calculated assuming the wire is infinitely long (B = I = 8 A).

Using the formula, we have percent difference = |(0.01 T - 8 A)/0.01 T| x 100% ≈ 799,900%.

Answer 2

To find the magnitude of the magnetic field B of the wire at a point on the y-axis, use the Biot-Savart law. The magnitude of B is 1.28 × 10^-5 T. The percent difference between the actual value and the value assuming the wire is infinitely long is 60%.

To calculate the magnitude of the magnetic field B of the wire at the point y=5.00 cm on the y-axis, we can use the Biot-Savart law. The formula is given by B = μ₀I/(2πr), where μ₀ is the permeability of free space, I is the current in the wire, and r is the distance from the wire. In this case, the wire is along the x-axis, so the distance from the wire to the point y=5.00 cm on the y-axis is 5.00 cm. Plugging in the values, we get B = (4π × 10^-7 T·m/A)(8.00 A)/[(2π)(0.050 m)] = 1.28 × 10^-5 T. Therefore, the magnitude of the magnetic field B is 1.28 × 10^-5 T.

Now, let's calculate the percent difference between the answer in part A and the value obtained by assuming the wire is infinitely long. According to the equation B = I/2πr, if the wire is infinitely long, we have B = (4π × 10^-7 T·m/A)(8.00 A)/(2π(0.050 m)) = 8.00 × 10^-6 T. The percent difference between the two values can be calculated by (|B_inf - B|/B_inf) × 100%, where B_inf is the value assuming the wire is infinitely long and B is the actual value obtained. Plugging in the values, we get (|8.00 × 10^-6 T - 1.28 × 10^-5 T|/8.00 × 10^-6 T) × 100% = 60%.