Two parallel long straight conductors lie on a smooth horizontal surface. Two other parallel conductor rest on them at right angles so as from B exits vertical. A uniform magnetic field B exists a vertical direction. Now all the four conductors stock moving outwards with a constant velocity v. The induced emf e and induced current i will vary wide time t as

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Answer

To solve the problem, we need to analyze the situation involving the four conductors, the magnetic field, and the motion of the conductors. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have two long straight conductors lying horizontally and two other conductors resting on them at right angles. A uniform magnetic field \( B \) is directed vertically, and all four conductors are moving outward with a constant velocity \( v \).

### Step 2: Determine the Change in Area
Initially, let the side length of the area formed by the conductors be \( a \). As the conductors move outward, the distance between them increases. After a time \( t \), each conductor moves a distance \( vt \). Therefore, the new side length \( a' \) becomes:
\[ a' = a + 2vt \]
The area \( A \) at any time \( t \) is given by:
\[ A = (a + 2vt)^2 \]

### Step 3: Calculate the Rate of Change of Area
To find the induced EMF, we need to calculate the rate of change of area with respect to time:
\[ \frac{dA}{dt} = \frac{d}{dt}[(a + 2vt)^2] = 2(a + 2vt) \cdot \frac{d}{dt}(a + 2vt) = 2(a + 2vt) \cdot 2v = 4v(a + 2vt) \]

### Step 4: Calculate the Rate of Change of Magnetic Flux
The magnetic flux \( \Phi \) through the area is given by:
\[ \Phi = B \cdot A \]
Thus, the rate of change of magnetic flux is:
\[ \frac{d\Phi}{dt} = B \cdot \frac{dA}{dt} = B \cdot 4v(a + 2vt) \]

### Step 5: Calculate the Induced EMF
According to Faraday's law of electromagnetic induction, the induced EMF \( e \) is equal to the negative rate of change of magnetic flux:
\[ e = -\frac{d\Phi}{dt} = -B \cdot 4v(a + 2vt) \]
Since we are interested in the magnitude, we can ignore the negative sign:
\[ e = 4Bv(a + 2vt) \]

### Step 6: Calculate the Induced Current
The resistance \( R \) of the conductors can be expressed in terms of their resistance per unit length \( r \) and the total length of the conductors. The total resistance \( R \) at any time is:
\[ R = r \cdot (4(a + 2vt)) \]
Thus, the induced current \( i \) can be calculated using Ohm's law \( i = \frac{e}{R} \):
\[ i = \frac{4Bv(a + 2vt)}{r \cdot (4(a + 2vt))} = \frac{Bv}{r} \]
This shows that the induced current \( i \) is constant over time since it does not depend on \( t \).

### Conclusion
- The induced EMF \( e \) increases with time as \( e = 4Bv(a + 2vt) \).
- The induced current \( i \) remains constant as \( i = \frac{Bv}{r} \).