To solve this problem, we need to analyze the forces acting on the metal rod when a magnetic field is applied. The rod is subject to gravitational force, frictional force, and the magnetic force due to the current flowing through it. Let's break this down step by step.

Understanding the Forces Involved

When the rod carries a current \( i \) and is placed in a magnetic field \( B \), it experiences a magnetic force given by the formula:

F_magnetic = i \cdot L \cdot B \cdot \sin(\theta)

where \( \theta \) is the angle between the direction of the current and the magnetic field. The gravitational force acting on the rod is:

F_gravity = m \cdot g

where \( g \) is the acceleration due to gravity. The frictional force that opposes the motion of the rod is given by:

F_friction = \mu \cdot F_normal

In this case, the normal force \( F_normal \) is equal to the gravitational force, so:

F_normal = m \cdot g

Thus, the frictional force becomes:

F_friction = \mu \cdot m \cdot g

Condition for Sliding

The rod will be on the verge of sliding when the magnetic force equals the maximum static frictional force. Therefore, we set the magnetic force equal to the frictional force:

i \cdot L \cdot B \cdot \sin(\theta) = \mu \cdot m \cdot g

Finding the Angle and Minimum Magnetic Field

To find the angle \( \theta \) that minimizes the magnetic field \( B \), we need to consider the sine function. The sine function reaches its maximum value of 1 when \( \theta = 90^\circ \). However, we are looking for the angle that allows the rod to be on the verge of sliding with the minimum magnetic field. This occurs when \( \sin(\theta) \) is maximized while still allowing for a balance of forces.

For practical purposes, we can analyze the scenario where the angle is less than \( 90^\circ \). The minimum magnetic field occurs when \( \theta \) is such that:

sin(θ) = 1

Thus, we can express the magnetic field as:

B = \frac{\mu \cdot m \cdot g}{i \cdot L}

Final Expressions

In summary, the angle \( \theta \) that puts the rod on the verge of sliding is:

θ = 90°

And the minimum magnetic field required to achieve this condition is:

B = \frac{\mu \cdot m \cdot g}{i \cdot L}

This analysis shows how the interplay of forces determines the conditions under which the rod will begin to slide, and how we can calculate the necessary magnetic field to achieve that state. Understanding these relationships is crucial in applications involving electromagnetism and friction.