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The lens equation is a crucial formula used to analyze the formation of images through lenses, particularly converging lenses. It is expressed as \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). This equation relates three important variables:

• \( f \): the focal length of the lens,
• \( d_o \): the distance from the object to the lens, known as the object distance,
• \( d_i \): the distance from the image to the lens, called the image distance.

By rearranging this equation, we can solve for any of these values if the other two are known. For instance, in a scenario involving a feather and two converging lenses, this equation helps determine how the feather's image is manipulated as it encounters each lens. Understanding how to use this equation is fundamental to grasping lens behavior.

The focal length, denoted as \( f \), is a measure of how strongly a lens focuses or diverges light. For a converging lens, this is the distance at which parallel rays of light converge to a single point. A shorter focal length implies a more powerful lens capable of bending light rays more steeply. In the context of our problem, both lenses have a focal length of 4.0 cm. This uniformity plays a significant role in manipulating the image as it passes through both lenses. Knowing the focal length is pivotal because it allows us to apply the lens equation effectively, helping us shape light paths and predict image characteristics.

Image distance, denoted as \( d_i \), is the distance from the lens to the formed image. It is one of the key parameters in the lens equation. When an object is placed at a certain distance from a lens, the image distance tells us where the image will appear relative to the lens. For the first lens in our problem, the calculated image distance, after solving the lens equation, is 6 cm. This means that the image created by the first lens is 6 cm away from it, forming on the side opposite to the incoming light. As we progress to the second lens, the position of this image becomes crucial, as it acts as the virtual object for the next stage of lens interaction.

Object distance, denoted as \( d_o \), is the distance from an object to the lens. It determines where the object is placed in relation to the lens before the image formation begins. In our exercise, the initial feather is placed 12 cm from the first lens. For a converging lens, correctly understanding the object distance is significant because it directly affects the image's characteristics such as size and position. As images progress through subsequent lenses, these distances readjust, leading to new object distances for the next lens stage. For instance, the intermediate image created by the first lens dictates the new object setup for the second lens, guiding us to solve for the separation between the lenses.