# Drawing Radial Field Equipotentials and Field Lines

• AN630078
In summary: Joules is a more conventional and easily recognizable unit. However, you are welcome to use kg⋅m2/s2 if that is more comfortable for you.

#### AN630078

Homework Statement
Hello, I have a question asking me to draw a radial gravitational field showing the gravitational field lines and the equipotential surfaces for equal energy increments. I then have to comment upon the spacing of the field lines and equipotential lines.

My query is in this question would it be referring to the spacing visually when drawing the field lines and equipotentials or what this spacing represents in a radial field?

I am not sure how to comprehensively answer what is being asked and would be very grateful for any advice
Relevant Equations
g=GM/r^2
I have just attached a standard depiction of a radial field as one may similarly choose to draw it. So I understand that the gravitational field strength in a field is defined as the force per unit mass at that point. The field lines in a radial field move further apart further away from the centre indicating the field strength is reducing. The closer together the field lines the stronger the field and thus the force.
The equipotential surfaces of a radial field are positions within a field with zero difference in potential between them; ie. the potential on an equipotential surface is the same everywhere as connected by equipotential lines. The field will always be perpendicular to the equipotential lines; since a field is defined as a region in which potential changes. How close the equipotentials are indicates the strength of the electric field and how quickly the potential is changing. e.g. a stronger field has closer equipotentials.

In a radial field, the field lines are all equally separated in terms of field strength but the radial distance between them increases as you move further from the planet. If this question is just asking how to draw a general radial field with field and equipotential lines then one would say that the field lines are evenly spaced but increase in distance further from the centre? And moreover, would the equipotential surfaces be said visually to be drawn unequally spaced with successive shells representing equal intervals of potential difference.

So for a radial field the field lines are evenly spaced but the equipotential surfaces are unevenly spaced?

Moreover, could an alternative unit for gravitational field strength besides N kg^-1 be Jm^-1kg^-1?

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You want equal energy increments so you have to start with the equation for the potential energy per unit mass, ##V(r)## not force per unit mass ##g(r)##.
If ##V(r)=-\dfrac{GM}{r}##, then $$\Delta V = GM\frac{\Delta r}{r^2}~\Rightarrow ~\frac{\Delta V}{V}=-\frac{\Delta r}{r}.$$Can you devise an iterative plotting procedure that exploits the above condition?

You can devise any alternative unit for gravitational field strength that you want, but m/s2 is more conventional and most easily recognizable by everyone.

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kuruman said:
You want equal energy increments so you have to start with the equation for the potential energy per unit mass, ##V(r)## not force per unit mass ##g(r)##.
If ##V(r)=-\dfrac{GM}{r}##, then $$\Delta V = GM\frac{\Delta r}{r^2}~\Rightarrow ~\frac{\Delta V}{V}=-\frac{\Delta r}{r}.$$Can you devise an iterative plotting procedure that exploits the above condition?

You can devise any alternative unit for gravitational field strength that you want, but m/s2 is more conventional and most easily recognizable by everyone.
Thank you for your reply. I do not know what you mean by an iterative plotting procedure.
In regard to an alternative unit for gravitational field strength, yes thank you for your suggestion I am aware of the use of m/s^2? However, would Jm^-1kg^-1 be a suitable alternative, although lesser used?

AN630078 said:
Thank you for your reply. I do not know what you mean by an iterative plotting procedure.
In regard to an alternative unit for gravitational field strength, yes thank you for your suggestion I am aware of the use of m/s^2? However, would Jm^-1kg^-1 be a suitable alternative, although lesser used?
Iterative procedure:
Decide on a value for the constant increment ##\Delta V##.
Find a number for the potential ##V## at ##r = R_E## (the Earth's radius) and draw an equipotential at that ##r##. Label it with the value of ##V.##
1. Solve the equation ##\dfrac{\Delta V}{V}=-\dfrac{\Delta r}{r}## to get the magnitude of the radius increment ##\Delta r## and add it to the old ##r##. Ignore the negative sign.
2. Add the increment ##\Delta V## to the old potential to get the new ##V## and draw an equipotential at the new ##r.## Label it with the new value of ##V.##
3. Go back to step 1 and repeat.
AN630078 said:
However, would Jm^-1kg^-1 be a suitable alternative, although lesser used?
In my opinion no, for the same reason that one uses Joules and not kg⋅m2/s2.

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## 1. What is the purpose of drawing radial field equipotentials and field lines?

The purpose of drawing radial field equipotentials and field lines is to visually represent the electric field in a given area. This can help in understanding the strength and direction of the electric field, as well as the distribution of electric potential.

## 2. How do you draw radial field equipotentials and field lines?

To draw radial field equipotentials and field lines, you first need to determine the location and strength of the electric charges in the area. Then, using the equations for electric field and electric potential, you can plot equipotential lines and field lines that are perpendicular to each other. The spacing between the lines represents the strength of the electric field.

## 3. What do the equipotential lines represent?

The equipotential lines represent points in the electric field that have the same electric potential. This means that no work is required to move a charge along an equipotential line, as the electric potential remains constant.

## 4. What do the field lines represent?

The field lines represent the direction and strength of the electric field. They point in the direction of the electric field and the closer the lines are together, the stronger the electric field is in that area.

## 5. Why are field lines perpendicular to equipotential lines?

This is because electric field and electric potential are related by the equation E = -∇V, where E is the electric field and V is the electric potential. This means that the electric field is always perpendicular to the equipotential lines, as the gradient of a scalar field is always perpendicular to the level curves of that field.