Tutor profile: Andrew V.
Subject: Physics (Waves and Optics)
We often handle lens/mirror problems that handle reflection and refraction. Our eyes contain lenses that focus light for our brains to process but over time degrade typically in the form of nearsightedness or farsightedness. What do these conditions physically mean for our eyes?
The best way to learn in my opinion is through meaningful examples so a question like this that begs for the Why's and How's instead of just the What's provide valuable insight and helps propel a student's motivation in wanting to understand the key concepts of any subject. When light travels into our eyes, it passes through three lenses: the tear film, cornea, and then the crystalline lens that stretches and contracts to focus on objects at different distances. Light passes through these and if all goes well, that light is focused perfectly onto the back of our eyes onto our retina where the information from incoming light can be sent to our brain to process an image. If a person is near sighted, the light that passes through our lenses is focused down to a point just short of our retina. This can happen for a few reasons. Either the crystalline lens in our eye is focusing light too strongly so that it converges far too soon, OR the eyeball itself might be stretched out so that the retina is too far back. A prescription for glasses to resolve this issue would contain minus lenses that would in essence weaken the focusing power of our eyes so that the focal point of light in our eye is pushed back onto the lens properly. But this begs another question. Why are nearby objects not focused too strongly too? Well, if we take in light from an object that is far away, the light rays that eventually make it to your eye end up entering nearly perpendicular to our lenses. These light rays will converge (come together to a point) too soon like we discussed and cause a blurry image. For an object nearby, the light from this object is still in the process of diverging (spreading outward) so when our eyes pick up that still diverging light, this ends up counterbalancing the nearsighted eye's attempts to make light converge too quickly! Someone who is farsighted has the exact opposite problem. The lenses in their eyes aren't focusing incoming light strongly enough so that the focal point ends up being behind the retina. This happens either when the lenses are too weak OR when the eyeball is too small, making the retina too close to the lenses. Light rays that enter the eye perpendicular to the lenses from far away objects will almost always be in focus, but once an object is up close, the crystalline lens is too rigid/inflexible to change its shape in order to refract light rays more strongly. We fix this issue by prescribing plus lenses that help the eye focus light even more. We often refer to these kinds of glasses as reading glasses since they are only needed to focus on objects that are up close! Understanding how the math behind lenses (and mirrors too) are crucial to being able to solve problems just like this one in far more detail so hopefully knowing at least one example as to why this knowledge is so useful is enough to spark an interest in the subject matter.
Subject: Physics (Electricity and Magnetism)
How does my microwave use electricity from an outlet to heat up my food?
I love this question because it pertains to an object whose ingenuity is often is taken for granted. When electricity comes from a typical wall outlet, that electricity is passed to a what we call a magnetron. A magnetron consists of a combination of many parts of which are most notable are a copper rod sitting within a grooved copper ring and our copper rod is often wrapped with a tungsten coil. The rod will serve as a negatively charged electrode (cathode) and our ring as a positively charged electrode (anode) so an electric field exists already between these pieces. When an electrical current flows through our tungsten coil, electrons are emitted towards our copper ring, generating a lot of heat. By applying a magnetic field to our rod/ring system, we make the electrons that leave the tungsten filament to begin to spiral outward from the rod to the sing instead of taking a straight path. How that magnetic field operates to spin the electrons is a whole other question that can be answered another time for the sake of keeping this answer somewhat short. This spiral motion creates what we can imagine to be a pinwheel of electrons spinning about the axis of the copper rod. The tips of this pinwheel pass over the grooves of the positively charged copper ring at regular intervals so that the inside of these grooves houses electromagnetic waves from the passing negatively charged pinwheel tips. Within these grooves are the microwaves themselves! An antenna can then guide the microwaves that we've generated into the chamber where our food resides. Once the microwaves are inside the chamber, bouncing off the walls of the microwave, they interact with the water molecules in the food. Water molecules are all polarized meaning that one end is positively charged while the other end is negatively charged. Since the microwaves form an electric field, the polarized water molecules will want to align with it. The microwaves are constantly bouncing around, hitting the food and walls, changing direction so that the electric field direction is always changing too and as a result, so does a water molecule. This increasing movement of our water molecules is what generates our heat!
What do derivatives and integrals mathematically mean and what are they used for in the real world?
The mathematical interpretation of a derivative is that the derivative of a function provides you with another function that maps out the slope of any given point along the original function. The most common example of derivatives being used in real life involves the motion of objects. Suppose we have a graph of an object's position vs. time that is generated with a position function. The slope of this graph can tell us about how fast or slow the object is moving. For example, if we have a large positive slope at one point in this graph, our object must be moving very fast at this point. By having the derivative of the position function, we would be able to determine the velocity that our moving object has at any point in time! Let's call this new function that we obtained by taking a derivative our velocity function. We can even take this a step further by taking a derivative of our velocity function. What this will do is create a new function that maps out the rate of change of our velocity which we call acceleration. We can call this new function our acceleration function. Both our velocity and acceleration functions can be plotted with respect to time and provide us with a wealth of information regarding the motion of the moving object. We can now determine the position, velocity, and acceleration of a given object at any point in time, all by starting with a simple position function! An integral on the other hand is, in a nutshell, the opposite of a derivative. An integral of a given function provides us with the area under the curve of said function. We can even use integrals to undo derivatives! We can tie this back into our previous example with derivatives starting with our acceleration function. Suppose we plotted our acceleration function with respect to time. If we were to take an integral of our acceleration function, the resulting function would be our velocity function! This is what I meant what I mentioned that integrals can "undo" a derivative. Taking this one final step further, the integral of our reclaimed velocity function would take us all the way back to our original position function! Neat, huh? There are a bevy of examples that we can walk through where derivatives and integrals are needed, but I believe this examples to be the most intuitive and easy to grasp.
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