Problem 73 - College Physics

Question

\(\bullet\) Three long, straight electrical cables, running north and south, are tightly enclosed in an insulating sheath. One of the cables carries a 23.0 A current southward; the other two carry currents of 17.5 \(\mathrm{A}\) and 11.3 A northward. Use Ampere's law to calculate the magnitude of the magnetic field at a distance of 10.0 \(\mathrm{m}\) from the cables.

Step-by-Step Solution

Answer

The magnetic field magnitude is \(1.16 \times 10^{-7} \, T\).

Step 1: Understand the context

We have three long, straight cables, each carrying a current. One cable carries a current southward, while the other two carry currents northward. We will use Ampere's Law to determine the magnetic field at a certain distance from these cables.

Step 2: List given values

The currents in the cables are: \( I_1 = 23.0 \, A \) southward, \( I_2 = 17.5 \, A \) northward, and \( I_3 = 11.3 \, A \) northward. The distance from the cables where we want to calculate the magnetic field is 10.0 meters.

Step 3: Calculate the net current using direction

Since current directions affect the net magnetic field, the southward current is considered negative and the northward currents are positive. Net current: \( I_{net} = (17.5 + 11.3 - 23.0) \, A = 5.8 \, A \).

Step 4: Apply Ampere's Law

Ampere's Law states that the line integral of the magnetic field around an amperian loop is equal to \( \, \mu_0 I_{enc} \, \), where \( I_{enc} \) is the net enclosed current. Using this, the magnetic field magnitude is \( B = \frac{\mu_0 I_{net}}{2 \pi r} \), where \( \mu_0 = 4 \pi \times 10^{-7} \, T\cdot m/A \) and \( r = 10.0 \, m \).

Step 5: Compute the magnetic field, B

Substitute the values into the formula: \( B = \frac{4 \pi \times 10^{-7} \times 5.8}{2 \pi \times 10} \, T \). Simplifying, \( B = \frac{2.32 \times 10^{-6}}{20} \, T = 1.16 \times 10^{-7} \, T \).

Step 6: Finalize the magnitude of B

The magnitude of the magnetic field at a distance of 10.0 meters from the cables is \( 1.16 \times 10^{-7} \, T \).

Key Concepts

Magnetic Field Calculation

Magnetic field calculation involves determining the magnetic influence created by electrical currents, particularly within current-carrying conductors. In our exercise, the focus is on calculating the resultant magnetic field 10 meters away from three parallel wires using Ampere's Law.

Ampere's Law states that the magnetic field around a closed loop is proportional to the electric current passing through the loop. Mathematically, this is \[ B = \frac{\mu_0 \cdot I_{net}}{2 \pi \cdot r} \]where \( \mu_0 \) is the permeability of free space (\( 4 \pi \times 10^{-7} \, T \cdot m/A \)), \( I_{net} \) is the net current, and \( r \) is the distance from the wire.

• Start by calculating the net current considering the direction of each current.
• Apply the condition of Ampere's Law in a loop passing through the cables.
• Solve for the magnetic field at the point of interest.

Using Ampere’s Law simplifies the calculations, especially in uniformly distributed current systems like ours.

Currents in Electrical Cables

Electrical cables may carry currents in various directions depending on the system design. Here, we have three cables: one cable carries a current of 23.0 A southward, while the other two carry currents of 17.5 A and 11.3 A northward.

When calculating effects like the magnetic field, the direction of current is crucial. For currents:

• Southward currents (opposite our reference direction) are taken as negative.
• Northward currents (along our reference direction) are positive.

Thus, these directions help determine the type and magnitude of the magnetic fields involved, further impacting the net current.

Net Current

The concept of net current is crucial in calculating magnetic fields using Ampere’s Law. The net current is the effective current responsible for producing the magnetic field and is calculated by vectorially adding the individual currents considering their directions.

In our scenario, you calculate net current \( I_{net} \) as follows:

• Sum the northward currents (positive values): \( 17.5 \, A + 11.3 \, A = 28.8 \, A \).
• Subtract the southward current (negative value): \( 28.8 \, A - 23.0 \, A = 5.8 \, A \).

The resulting net current of 5.8 A points northward. This value is then used in Ampere's Law to find the resulting magnetic field.

Magnetic Field Magnitude

Magnetic field magnitude expresses the strength of the magnetic field created by the system of currents. To find this, we use Ampere’s Law and the net current calculated earlier.

Using the formula \[ B = \frac{\mu_0 \cdot I_{net}}{2 \pi \cdot r} \],substitute:

• \( \mu_0 = 4 \pi \times 10^{-7} \, T \cdot m/A \)
• \( I_{net} = 5.8 \, A \)
• and \( r = 10.0 \, m \)

Plug these into the equation:
\[ B = \frac{4 \pi \times 10^{-7} \, T \cdot m/A \times 5.8 \, A}{2 \pi \times 10.0 \, m} \]
The computed magnitude becomes \( 1.16 \times 10^{-7} \, T \). This magnetic field magnitude shows the influence of the currents at the specified distance.