Question

At what rate a single conductor should cut the magnetic flux so that current of $1.5\ \text{mA}$ flows through it when a resistance of $5\ \Omega$ is connected across its ends?

• A. $6 \times 10^{-3}\ \text{wb s}^{-1}$
• B. $8 \times 10^{-3}\ \text{wb s}^{-1}$
• C. $4 \times 10^{-4}\ \text{wb s}^{-1}$
• D. $7.5 \times 10^{-3}\ \text{wb s}^{-1}$

Hint:

Remember that $1\ \text{Volt}$ is equivalent to $1\ \text{Weber per second}$ ($\text{Wb s}^{-1}$). Finding the rate of flux cutting is physically identical to calculating the target open-circuit voltage drop across the system using a direct Ohm's Law calculation ($I \times R$).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem asks for the rate at which a conductor must cut across magnetic flux lines ($\frac{d\phi}{dt}$) to generate a target electric current ($I$) through a circuit containing a known localized electrical resistance ($R$).

Step 2: Key Formula or Approach:
1. According to Ohm's Law, the induced e.m.f. ($e$) needed to drive a current $I$ through a loop of resistance $R$ is:
$$e = I \cdot R$$ 2. Faraday's Law states that the induced e.m.f. matches the rate of change of magnetic flux cutting across the conductor:
$$e = \frac{d\phi}{dt}$$ Combining these two baseline formulas links the variables directly: $\frac{d\phi}{dt} = I \cdot R$.

Step 3: Detailed Explanation:
Let's convert the given values into standard SI units:
Current, $I = 1.5\ \text{mA} = 1.5 \times 10^{-3}\ \text{A}$ Resistance, $R = 5\ \Omega$ Equate the expression from Faraday's law to Ohm's law to solve for the flux cutting rate:
$$\frac{d\phi}{dt} = I \cdot R$$ Substitute our values directly into this equation:
$$\frac{d\phi}{dt} = \left(1.5 \times 10^{-3}\ \text{A}\right) \times 5\ \Omega$$ Multiply the coefficients together:
$$\frac{d\phi}{dt} = 7.5 \times 10^{-3}\ \text{wb s}^{-1}$$ This calculation determines the required rate of change of magnetic flux in Webers per second.

Step 4: Final Answer:
The conductor must cut magnetic flux at a rate of $7.5 \times 10^{-3}\ \text{wb s}^{-1}$, corresponding to option (D).