A metallic ring of field. If z is the radius l (ring being horizontal is falling under gravity in a region haivng a magnetic field. If z is the vertical direction, the z-component of magnetic field is `B_(z) = B_(0) (1 + lambda z)`. If R the resistance of the ring and if the ring falls with a velocity `upsilon`, find the energy lost in the resistance If the ring has reached a constant velocity, use the conservation of energy to determine `upsilon` in terms of m, B, `lambda` and acceleration due to gravity g.

Explanation

To solve the problem step by step, we will analyze the situation involving a metallic ring falling under the influence of gravity in a magnetic field that varies with height. The magnetic field is given by \( B_z = B_0 (1 + \lambda z) \). We will find the energy lost in the resistance of the ring and determine the constant velocity \( \upsilon \) in terms of mass \( m \), magnetic field \( B \), \( \lambda \), and gravitational acceleration \( g \).

### Step 1: Determine the Induced EMF
The induced electromotive force (EMF) in the ring can be calculated using Faraday's law of electromagnetic induction. The EMF \( \mathcal{E} \) is given by:

\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
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