How do math geniuses understand extremely hard math concepts so quickly?
Some years ago I was doing a penetration test for a large mobile operator's voicemail infrastructure. The test in itself took about 5 minutes from the moment I started till the moment I was able to get into the system and change the default welcome message to something of my own. I presented my findings to the local manager in charge and, surprisingly, her reaction was - "so you expect us to pay you 2000€ for 5 minutes of work?" My answer, although a bit of a cliche, was this: "You're not paying for those 5 minutes. You're paying for the amount of knowledge gathered over the years that allowed me to figure out in less than a minute what's the problem with your voicemail." As Satvik Beri perfectly described in his answer, when you spend most of your time focusing on a specific issue, whether that's math, information security, programming or anything else - you gather enough experience to start figuring out things in a snap. That doesn't make you special, nor innately talented - just a really hard worker and extremely passionate about what you do. Think about that the next time you feel cheated by a consultant who solves your problem in 5 minutes.
Some conceptual questions regarding equal potential fields. I think I understand them but I'm not sure?
(Positive probe) <---- Electric Field in water ----> (Negative probe) 1. What happens to a negative charge as it moves from the positive test probe to the negative bar? 2. What could cause a negative charge to move from the positive test probe to the negative bar? 3. Where would a positive test charge have the least potential energy?
The relationship and differences between electric potential and electric potential energy.....................?
In the gravitational world: F(z) = m*g g is the gravitational field. It already contains a (-ž) in its direction, because it is a vector quantity. Force of gravity in a uniform gravitational field does not depend on location. The gravitational potential is given by: GP = -g·h g·h represents the dot product of g and h. It is negative, because g points downward, and h points upward. We want the scalar quantity to be positive when h is positive. The gravitational potential energy is given by: GPE = m*(-g·h) ----------------------------- In the electric world: Force: F, a vector quantity pointing in the direction of the force on our charge of interest Electric field: E, a vector quantity which is by definition, E = Fnet/q, where q is the charge of interest placed at a particular location. Electric potential: V. A scalar quantity representing the work required per unit charge, to bring a charge to the location of interest, from the datum, the location defined as V=0. Definition: V = -∫E·dx. Electric potential energy: EPE (I hate calling it U): EPE is a scalar quantity representing the work required to bring a particular charge of interest to the location of interest, from the datum. ----- Critical to understand: Electric field (E) only depends on the pre-existing arrangement of charges, presuming no magnetism effects exist. It doesn't depend on what charge is placed at the location, and it always points in the direction a possible positive charge will be directed by the force. Electric potential (V), often called Voltage, is like electric field, in that it only depends on the pre-existing arrangement of charges (and also the choice of datum). Both electric net force Fnet and electric potential energy EPE depend on the charge of interest chosen to be placed at the location of interest. They are related to electric field and electric potential as follows: Fnet = q*E EPE = q*V
What is the relationship between electric field strength and the potential gradient?
The electric field is the negative of the gradient of potential. E = -grad V. It points in the direction of steepest descent of the potential. Yes, it is a rate of change but when you are dealing with more than one dimension, there is more than one rate of change. You can go a long way with the height analogy for potential. Think of the potential everywhere in space as the height above the ground. If you are on a hill, there is more than one uphill direction. But only one of them, the steepest ascent direction, is the one called "gradient". And the negative gradient is the steepest descent. Electric field is force per unit charge. The steeper the descent, the more the force pushing the charge down the hill. Just as with height and masses. The main difference between charge and mass in this analogy is that there is such a thing as negative charge (electrons). And they fall "uphill". Edit: Another fact that may come in handy: If you have a contour plot of potential (or height), the steepest ascent/descent directions are perpendicular to those contours.
Electricity question (Concept)?
I think I'll answer your second question first: "Voltage" is another term for "Potential Difference". Now, by definition, on an equipotential line, the potential is the same for all points. So the voltage between two points on an equipotential line is the same as the potential difference between two points on the line, which by definition is 0 because their potentials are the same! Now for the first part: It is actually possible for equipotential lines to intersect - but there is a strong condition for that: The potentials on the lines crossing must be the same. See below to understand why: Suppose two equipotential lines, L1 and L2, cross at point X, and at point X they have some potential P. Because L1 is an equipotential line that contains point X, all points on L1 must have potential P. But also, L2 is an equipotential line that contains point X, and so all points on L2 must have potential P. Therefore the two equipotential lines L1 and L2 have the same potential everywhere. A common way for this crossing to occur is in what is known as a "saddle point" in the electric field. How easy this is to generate (if possible), I'm afraid I don't know, but it is possible. (I'm not really sure what to do with your 3rd question, but I can say that if you are looking at a graph with equipotentials on it (like contours on a map), equipotentials start forming smaller and smaller loops when they get near maxima and minima.) Hope this helps :)
Do Affirmative Action policies discriminate against Whites?
Actually a friend of mine worked on the "Workplace 2000" and "Workplace 2020" book series and they (a Conservative think tank) came out in support of Affirmative Action (to a degree). In their studies Affirmative Action is proven to be a correction for past actions. It's benefit is readily noticable. It is noticable in future generations of the families. The family that does better financially in this generation provides for better education for the next generation. In several generations it provides for a better-educated work-force, which is better for everyone concerned. Oh, and the concept of reverse-racism or "discrimination against white" is only possible in a setting where white are no longer the plurality (note, I didn't say majority). Thus, no: there is nothing discriminatory against white people. Those who claim such have limited understanding of ecomomics, etc. The concept is to increase the size of the pie, not to force the same pie into smaller slices. And if you are in a 1 to 50 ratio of applications to jobs and you're blaming on something else, I know why you didn't get hired.
Questions about escape velocity?
I'm to understand that for an object to achieve escape velocity, its kinetic energy has to be equal to that of its gravitational potential energy. Unfortunately, the mechanics of motion while under an opposing force such as gravity has always confused me. If both forces of KE and gpe are equal, why exactly should the object move at all? Why wouldn't it just hover weightless or something? I'm sorry it's a little vague, but could someone help me visualize this concept better?