For many problems about electric fields and forces it is important to calculate the whole vector. By the whole vector I mean the x-component and the y-component of the vector. Knowing these two numbers, which can be negative or positive, means we know everything about the vector (if you work in three dimension you will also need the z-component). When there are more than one source of an electric field you need to add the electric field vector produced by the one source to the electric field vector produced by the other source. The way you add vectors is by adding each kind of component separately, that is: Figure: The x-component of the total E-field at certain point is the x-component of E of the first source plus the x-component of E of the second source, and similarly for the y-component. So in order to be able to add the x-components and y-components we obviously first need to know what the x- and y- components are.

Strategy

There is a simple strategy that gives us the x- and y-components of electric fields directly rather than having to go through first calculating the magnitude of the electric field: 1. First you calculates the x- and y-components of the distance from the source to the target (rx and ry), and the magnitude of that distance. We need to be careful about signs when it comes to rx and ry. They are defined as how far you have to walk only in x- or y-direction in order to get from the source to the target. For example, if you have to walk to the right in x-direction that means rx is positive; if you have to walk to the left in x-direction to get from the source to the target rx will be negative. Similarly up and down are positive and negative, respectively, for the y-direction.

Example: 2. Now we insert the values we got for rx, ry and r  into the following equations: Figure: The first part after the equal sign has the same magnitude as E. However, in order to get to the X component of E we only need a fraction of E (since Ex is always smaller than E). That’s why we multiply E by a fraction rx/r, which is equal to the fraction that Ex/E. There are two possible places where a negative sign can come in when calculating Ex: The charge and rx and ry. So make sure you always put in the correct sign. This way, the signs for Ex and Ey will automatically come out correctly. Example: Say we use the above situation and with Qs = -1 C. What is the E-field at the source.

Example:

From intuition we know about a property that many objects have: mass. Another propoerty that many particles have is called electric charge. The reason we don’t have a good inuition for this property is that there are two kinds, positive and negative charges and they can and do cancel each other out. In everyday object the number of positive charges and negative charges are ver finely balanced so that the overall charge is zero. Only sometimes do we notice a surplus of positive or negative charges when we get a shock touching a car door, or when we hear and see sparks, when unfolding laundry freshly out of the dryer, for example.

Just like the mass of an object creating gravity, the charge of a n object changes the environment everywher in space. It makes its presence known, so to speak. When another charge is placed somewhere else in space, this second charge will “see” the new environment at the place where the second charge resides and in response feel a force. Again, this is similar to gravity where the Earth, for example, changes the environment everywhere, so also at the place where we exist. Then, we feel a force (gravity) because we interact with this new environment. While for the mass property the new environment is called gravity, for charges it is called the Electric Field.

Every charge creates such an electric field. However, sometimes where not interested in the electric field created by a certain charge. For example if we ask the question what is the force on charge 1 by charge 2, we only care about the electric field produced by charge 2 at the position of charge 1. This is the electric field that charge 2 needs to interact with, in order to feel a force. We do not care what the electric field is, which is produced by charge 1 itself  because it cannot produce a force on itself. This is similar to how you cannot pull yourself up by your own hair.

If have a problem like “what is the force on charge 1 by charge 2?” we can therefore label charge 2 the source of the electric field and charge 1 the target of the electric field. This way it’s clear which charge is producing the electric field that we’re interested in (source) and which charge reacts to the electric field and feels a force (target). Of course both charges still produce an electric field, but we only need to know about the electric field produced by the source.

Figure: A question like this one automatically turns Q1 into the target and Q2 into the source. This means we care about Q2 in terms of what electric field it creates and about Q1 in terms of how much force it feels in response to the electric field created by the source.

Strategy:

Always put units into their standard form before beginning a problem. Every physical quantity, like mass, force or distance, has a universally accepted standard unit which makes conversion from one number for mass into another for distance easy. The most important standard units are:

• Mass – kilogram (kg)
• Distance – meter (m)
• Time – second (s)
• Force – Newton (N)
• Energy – Joule (J)
• Power – Watt
• Electric Charge – Coulomb (C)
• Magnetic Field – Tesla (T)

Sometimes numbers are given in different units that are fractions or multiples of these units. For example, 1/100 of a meter. There are abbreviations for fractions and multiples like this. The most important ones are:

• 1/100 – centi (c)
• 1/1000 – milli (m)
• 1/(10^6) – micro (u)
• 1/(10^9) – nano (n)
• 1000 – kilo (k)
• 10^6 – Mega (M)
• 10^9 – Giga (G)

When a number is given with such an abbreviation, just replace the abbreviation with the actual number it represents. For example 5 cm is 5*10^2 m.

Strategy:

One of the most basic, common and important tools for solving physics problems is to actually solve an equation once you’ve completed all the physics. For linear equations with one unknown you should basically follow these steps:

1 . Get all x to the top

2. Get rid of brackets by factoring out

3. Bring all x to one side and all single numbers to the other

4. Isolate x by factoring, leaving only one x

5. Completely isolate x by dividing out numbers

Sometimes some of these steps are not necessary, but no matter what equation you get these should be all you ever have to do to solve such an equation.

Example:

Practice problems:

1. (x-7)+(6x-2) = 25

2. 3+7x/(5+x) = -5

3. 10x-(3x+5)-10 = 4x

Answers:

1. x = 34/7

2. x = -2

3. x = 5