Problem 56 - Algebra 2

Question

MAGNETS For a bar magnet, the magnetic field strength \(H\) at a point \(P\) along the axis of the magnet is \(H=\frac{m}{2 L(d-L)^{2}}-\frac{m}{2 L(d+L)^{2}}\) . Write a simpler expression for \(H .\)

Step-by-Step Solution

Answer

The simplified expression for the magnetic field strength is \(H = \frac{2md}{(d-L)^2(d+L)^2}\).

Step 1: Understand the Expression

We begin with the expression: \(H = \frac{m}{2L(d-L)^2} - \frac{m}{2L(d+L)^2}\). We need to simplify it.

Step 2: Identify a Common Denominator

The fractions in the expression have different denominators, but we can combine them by finding a common denominator. The common denominator is \(2L(d-L)^2(d+L)^2\).

Step 3: Rewrite Fractions with Common Denominator

Rewrite each fraction with the common denominator: \(H = \frac{m(d+L)^2}{2L(d-L)^2(d+L)^2} - \frac{m(d-L)^2}{2L(d-L)^2(d+L)^2}\).

Step 4: Combine the Fractions

Combine the fractions under the common denominator: \(H = \frac{m(d+L)^2 - m(d-L)^2}{2L(d-L)^2(d+L)^2}\).

Step 5: Expand and Simplify the Numerator

Expand the numerator \(m[(d+L)^2 - (d-L)^2]\) to get \(m[(d^2 + 2dL + L^2) - (d^2 - 2dL + L^2)] = m[4dL]\).

Step 6: Final Simplification

Substitute the simplified numerator back into the expression: \(H = \frac{4mdL}{2L(d-L)^2(d+L)^2}\). Simplify to \(H = \frac{2md}{(d-L)^2(d+L)^2}\) by canceling \(2L\) across the numerator and denominator where possible.

Key Concepts

Common Denominator

When dealing with fractions, especially in algebraic expressions, finding a common denominator is crucial to simplify or combine them easily. Imagine you are adding or subtracting fractions; each fraction must have the same base (denominator) to ensure a smooth operation. In the case of our expression \(H = \frac{m}{2L(d-L)^2} - \frac{m}{2L(d+L)^2}\), both fractions have denominators involving different quadratic expressions.

To simplify, we find the common denominator here: \(2L(d-L)^2(d+L)^2\). This allows us to rewrite each fraction over this shared foundation.

• Write each numerator with respect to the common denominator.
• This often involves multiplying the existing numerator by whatever factors are missing in the leader fraction's denominator to make it complete.

By doing this, we ensure that operations like addition or subtraction of these fractions are streamlined.

Fraction Simplification

Fraction simplification is a method to make expressions more manageable by reducing the complexity of both the numerator and the denominator. After establishing a common denominator, simplifying the fractions follows.

Once we have our expression rewritten as \(H = \frac{m(d+L)^2}{2L(d-L)^2(d+L)^2} - \frac{m(d-L)^2}{2L(d-L)^2(d+L)^2}\), the next step is to combine them since they share the same denominator.

• This results in a single fraction: \(H = \frac{m(d+L)^2 - m(d-L)^2}{2L(d-L)^2(d+L)^2}\).
• The key to simplification lies in minimizing common terms in both numerators and denominators through cancellation or arithmetic operations.

Simplification not only makes expressions compact but also prepares them for any further mathematical manipulations that may be required in problem-solving fields such as physics or engineering.

Numerator Expansion

Expanding the numerator is a critical step to simplify complex fractions. Here, by expanding the terms, we open up opportunities to eliminate or reduce repeated terms. In our expression \(H = \frac{m(d+L)^2 - m(d-L)^2}{2L(d-L)^2(d+L)^2}\), the focus is on the expressions within the numerator.

We first address \((d+L)^2 - (d-L)^2\) using the formula for expanded squares, which is \((x+y)^2 = x^2 + 2xy + y^2\) and \((x-y)^2 = x^2 - 2xy + y^2\).

• Substituting these into the formula delves into the benefits of eliminating terms: yields \(m[4dL]\).
• By expanding, simplifying, and canceling unwanted terms, it's clearer and easier to manage.

This expanded form is now ready for reduction and further simplification, leading us to the final result of \(H = \frac{2md}{(d-L)^2(d+L)^2}\). Thus, numerator expansion plays a pivotal role in algebraically manipulating complex expressions efficiently.