A 2.0 cm×2.0 cm square loop of wire with resistance 0.010 Ω has one edge parallel to a long straight wire. The near edge of the loop is 1.0 cm from the wire. The current in the wire is increasing at the rate of 100 A/s. What is the current in the loop?

Verified step by step guidance

1. Step 1: Understand the problem setup. The square loop of wire is placed near a long straight wire, and the current in the straight wire is increasing at a rate of 100 A/s. This changing current creates a changing magnetic field, which induces an electromotive force (EMF) in the loop according to Faraday's Law of Induction.
2. Step 2: Calculate the magnetic flux through the loop. The magnetic field due to the straight wire at a distance r is given by \( B = \frac{\mu_0 I}{2 \pi r} \), where \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \)), \( I \) is the current in the wire, and \( r \) is the distance from the wire. Integrate this magnetic field over the area of the loop to find the total flux.
3. Step 3: Use Faraday's Law to find the induced EMF. Faraday's Law states \( \text{EMF} = -\frac{d\Phi_B}{dt} \), where \( \Phi_B \) is the magnetic flux. Since the current in the wire is increasing at a rate of 100 A/s, the rate of change of the magnetic flux can be calculated using the derivative of the flux with respect to time.
4. Step 4: Relate the induced EMF to the current in the loop. The induced current in the loop can be found using Ohm's Law: \( I = \frac{\text{EMF}}{R} \), where \( R \) is the resistance of the loop (given as 0.010 Ω). Substitute the value of the EMF calculated in the previous step into this equation.
5. Step 5: Combine all the expressions and substitute the given values. Use the dimensions of the loop (2.0 cm × 2.0 cm), the distance from the wire (1.0 cm), the rate of change of current in the wire (100 A/s), and the resistance of the loop (0.010 Ω) to compute the induced current. Ensure all units are consistent (convert cm to meters where necessary).

Key Concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop. According to Faraday's law, the induced EMF is proportional to the rate of change of magnetic flux through the loop. In this scenario, the increasing current in the straight wire generates a changing magnetic field that affects the square loop.

Magnetic Field due to a Long Straight Wire

The magnetic field generated by a long straight wire carrying current can be calculated using Ampère's law. The magnetic field (B) at a distance (r) from the wire is given by the formula B = (μ₀I)/(2πr), where μ₀ is the permeability of free space and I is the current. This magnetic field influences the loop, and its variation as the current changes is crucial for determining the induced current in the loop.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor, expressed as I = V/R. In this problem, once the induced EMF in the loop is calculated, Ohm's Law will be used to find the current flowing through the loop based on its resistance.