This page demonstrates mathematically how a parabolic antenna works. Parallel electromagnetic rays along the parabola's axis are reflected to the focus. Conversely, waves emitted from the focus reflect off the parabola and propagate parallel to the axis.
An electromagnetic ray parallel to the parabola's axis (the \(y\)-axis) hits the inner surface of the parabola (see figure below).
Let \(M(a,b)\) be the point where the ray hits the parabola. Let \(i\) be the angle of incidence with respect to the normal, and \(r\) the angle of reflection. According to the law of reflection, \(i = r\). We will show that all reflected rays intersect the parabola's axis at the same point.
The parabola equation is: \[ y = \frac{x^2}{4f} \] where \(f\) is the focal distance. The derivative is: \[ y' = \frac{x}{2f} \] giving the tangent slope at \(M(a,b)\): \[ m_t = \frac{a}{2f}. \] The slope of the normal satisfies: \[ m_t \cdot m_n = -1 \quad \Rightarrow \quad m_n = -\frac{2f}{a}. \] Let \(n\) be the angle of the normal with the \(x\)-axis: \[ \tan(n) = -\frac{2f}{a}. \] The slope of the reflected ray is: \[ m_r = \tan(n - i) \quad \text{and} \quad i + n = 90^\circ. \] Using trigonometric identities: \[ m_r = \tan(2n - 90^\circ) = -\cot(2n) = -\frac{1}{\tan(2n)} = -\frac{1 - \tan^2 n}{2 \tan n}. \] Substitute \(\tan n = -2f/a\): \[ m_r = \frac{a^2 - 4f^2}{4af}. \] The line through \(M(a,b)\) with slope \(m_r\) is: \[ y - b = m_r (x - a). \] The y-intercept is: \[ y_\text{intercept} = b - a m_r = \frac{a^2}{4f} - a \frac{a^2 - 4f^2}{4af} = f. \] Thus, all reflected rays pass through the focus \((0,f)\).
Note: This derivation uses algebra, trigonometry, calculus, and physics, demonstrating the real-world application of mathematics in engineering and physics.
Incoming electromagnetic rays reflect off the parabolic surface and converge at the focus.
Rays emitted from the focus reflect off the parabola and travel parallel to the axis.
Find the focus of each parabola: