A conductor is moving in the magnetic field B the induced current is I. if the magnetic field is doubled, the induced current will

Options

A. remain the same
B. be half
C. be double
D. be four times

Explanation

To solve the problem, we need to understand the relationship between the induced current, the magnetic field, and the electromotive force (emf) generated in the conductor. Here’s a step-by-step solution:

### Step 1: Understand the relationship between magnetic field and induced emf
When a conductor moves through a magnetic field, an electromotive force (emf) is induced in the conductor. The induced emf (ε) is given by Faraday's law of electromagnetic induction, which states that:
\[ \varepsilon = -\frac{d\Phi}{dt} \]
where \(\Phi\) is the magnetic flux.

### Step 2: Define magnetic flux
Magnetic flux (\(\Phi\)) is defined as:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where \(B\) is the magnetic field strength, \(A\) is the area through which the magnetic field lines pass, and \(\theta\) is the angle between the magnetic field and the normal to the surface area \(A\). For simplicity, we can assume \(\theta = 0\) (the magnetic field is perpendicular to the area), so:
\[ \Phi = B \cdot A \]

### Step 3: Analyze the effect of doubling the magnetic field
If the magnetic field \(B\) is doubled, the new magnetic flux \(\Phi'\) becomes:
\[ \Phi' = 2B \cdot A \]
This means that the change in magnetic flux (\(d\Phi\)) over time will also be doubled, assuming the area \(A\) and the motion of the conductor remain the same.

### Step 4: Relate induced emf to induced current
Since the induced emf is directly proportional to the change in magnetic flux:
\[ \varepsilon' = -\frac{d\Phi'}{dt} = -\frac{2d\Phi}{dt} = 2\varepsilon \]
This indicates that the induced emf is now doubled.

### Step 5: Apply Ohm's Law
According to Ohm's Law, the relationship between voltage (emf), current (I), and resistance (R) is given by:
\[ V = I \cdot R \]
If the emf is doubled and the resistance remains constant, the new current \(I'\) can be expressed as:
\[ I' = \frac{\varepsilon'}{R} = \frac{2\varepsilon}{R} = 2I \]
Thus, the induced current will also be doubled.

### Conclusion
If the magnetic field is doubled, the induced current will also be doubled.