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Whether or not to run in the rain

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2012 Eur. J. Phys. 33 1321

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IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 33 (2012) 1321–1332 doi:10.1088/0143-0807/33/5/1321

Whether or not to run in the rain

Franco Bocci

Department of Mechanical and Industrial Engineering, University of Brescia, via Branze 38,

Brescia, Italy

E-mail: franco.bocci@ing.unibs.it

Received 28 March 2012, in ﬁnal form 26 June 2012

Published 19 July 2012

Online at stacks.iop.org/EJP/33/1321

Abstract

The problem of choosing an optimal strategy for moving in the rain has attracted

considerable attention among physicists and other scientists. Taking a novel

approach, this paper shows, by studying simple shaped bodies, that the answer

depends on the shape and orientation of the moving body and on wind direction

and intensity. For different body shapes, the best strategy may be different: in

some cases, it is best to run as fast as possible, while in some others there is an

optimal speed.

(Some ﬁgures may appear in colour only in the online journal)

1. Introduction

Is it better to walk or run in the rain? Almost everyone has at least once faced this question; many

discussions on this topic can be found on the web, with various conclusions. The problem has

also been addressed by some physicists, mathematicians, engineers and meteorologists. Some

studies [1] consider vertically falling rain. Others [2,3] take into account the possibility of wind

with the same direction of motion. Others still [4–8] also consider a cross (i.e. perpendicular

to the path) component of the wind. Since the human body has a very complex shape, a

much simpler form, usually a parallelepiped, has been considered, with the implicit or explicit

assumption that shape is not a crucial factor, and that results obtained for a parallelepiped can

be quite easily generalized to bodies of any shape.

Qualitatively speaking, the results found up to now can be summarized as follows.

•For rain falling vertically, the best strategy is to run as quickly as possible. The same is

also true for motion into the wind.

•For motion downwind, there may be an optimal speed, which equals the component along

the direction of motion of the wind velocity. This happens only if the ratio between the

cross-section of the body perpendicular to the motion and the horizontal one is large

enough; otherwise, the best choice is again to run at the maximum speed one can reach.

In essence, in the literature so far published on this subject there are two main conclusions:

for the existence of an optimal speed the wind must come from behind and the optimal speed,

when it exists, equals the component of the wind velocity along the path. In this paper, we

0143-0807/12/051321+12$33.00 c

2012 IOP Publishing Ltd Printed in the UK & the USA 1321

1322 F Bocci

shall show that the answer to the question actually depends on many factors, mainly on the

shape of the body and its orientation. Even though a tailwind is a favourable condition for the

existence of an optimal speed, in some cases we can have one even with a headwind, and in

general its value is not equal to that found in previous studies.

The paper is addressed mainly to undergraduate students and their teachers, as well as to

anyone intrigued by the problem. The subject requires good geometrical visualization, a good

command of vector calculus and involves many concepts from various ﬁelds; as such, it could

be useful in physics courses at university level.

2. Deﬁnition of the problem

The question stated at the very beginning of this paper is too vague, and should be better

deﬁned.

Let us consider a ﬁxed track that has to be covered in the rain. We shall assume the

following.

(1) The ground is horizontal.

(2) The path is rectilinear.

(3) The rain is uniform, at least along the path.

(4) The rain is constant, at least for the time needed to cover the path.

(5) The body motion is rigid and translatory.

This last assumption is clearly a quite crude approximation of a human body, but it is

necessary to make the problem manageable. The conditions (2), (3) and (4) are not as restrictive

as they may look at ﬁrst sight: if they are not satisﬁed on the whole path, we can always split

it into shorter parts.

3. Introduction of conceptual tools

Let us suppose that we have to go straight from point A to point B. We choose a Cartesian

reference system with the xaxis oriented from A to B, and the zaxis vertical. Let ube our

velocity; in this frame, u=(u,0,0), with u>0. The problem is to ﬁnd the value of u,ifany,

that will minimize the mass of water impinging on our body along the path.

Let v=(vx,v

y,v

z)be the rain velocity. The vertical component, vz, depends on the drop

size. We shall assume that drops fall at their terminal speed, so that their horizontal velocity

equals that of the wind (we do not restrict ourselves to a wind along the xdirection, but we

assume a horizontal wind). The sign of vysimply tells us if we get wet on our left or right side.

When vxis positive, we shall speak of a tailwind, otherwise of a headwind, irrespective of the

presence of a cross component vy.

The plane identiﬁed by uand vplays an important role in the problem: we shall call it π;

an axis perpendicular to this plane will be denoted by ζ.

Let us call ρthe ratio between the mass of water drops that are found, at some instant,

within a given volume, and the volume itself (note that ρis not the water density). We can

now deﬁne a vector j0that we shall call rain density:

j0=ρv.(1)

It is easy to recognize the analogy between ρand charge density, and between j0and the

current density vector in electromagnetism. Our approach will freely exploit conceptual tools

taken from this ﬁeld. Assumptions (3) and (4) imply ρand v—and consequently j0—to be

uniform and constant.

Whether or not to run in the rain 1323

x

y

z

u

v j

plane

Figure 1. The relative orientation of v,u,jand the plane π, which contains all of them.

When moving, we shall perceive an apparent rain density j, which will differ from j0:

j(u)=ρ(v−u)=ρ(vx−u,v

y,v

z). (2)

The jvector changes, in intensity and direction, with u. Its direction rotates, when u

increases, in the plane π, drifting away from the positive xdirection; we can easily see this

effect on train or car windows. In astronomy, this effect is known as stellar aberration. Sailors

know well that the direction from which the apparent wind comes approaches the bow as boat

speed increases. We can state this concept more precisely by saying that the angle θbetween

jand uincreases with u, approaching 180◦. It is easy to show that

tan θ=v2

y+v2

z

vx−u.(3)

We can note that j0is the value of jcorresponding to u=0; we shall call it the initial

value of j. Similarly, the angle between j0and uwill be denoted by θ0.

Figure 1shows the relative orientation of v,u,j, and the plane π.

Using once more the analogy with electromagnetism, it is natural at this point to introduce

the concept of rain ﬂux. If we adopted without any change the ﬂux deﬁnition for a vector ﬁeld,

we should deﬁne rain ﬂux as the surface integral of the jﬁeld:

(u)=S

j·dA,(4)

where the integration is over the whole surface of the body. But we know that, in the absence

of ﬁeld sources or wells, the integral over a closed surface vanishes: thus the integral (4) must

be modiﬁed for our purposes.

A ﬁrst change is to consider the modulus of the dot product j·dA. But this is not

satisfactory, as every ﬁeld line contributes twice to the ﬂux: when it enters the body and when

it leaves. So we shall restrict the integration to what we shall call the ‘wet surface’ of our body,

Sw, that is the surface where ﬁeld lines enter the body. If the body has a complex shape, so that

there are some ﬁeld lines that enter, go out and enter again, the wet surface obviously includes

only those parts where ﬁeld lines enter the body for the ﬁrst time. We then deﬁne rain ﬂux as:

(u)=Sw

|j·dA|.(5)

Note that, while Sis a closed surface, Swis always an open one, as the different notation

of the integral remind us. We shall assume that it maintains the same orientation as S(which,

by convention, is oriented from inside to outside).

1324 F Bocci

Rain ﬂux is the ratio between the mass of water impinging on the body during a given

time interval and the time interval; its SI measure unit will be kg s−1. It depends on the body

speed for two reasons, because jchanges with uand because, due to the rotation of j,the

surface of integration may change.

4. Formalisation and general discussion of the problem

We can now, at least in theory, calculate the water mass, m, hitting the body during motion in

the rain. The time interval, t, needed to travel a distance Lat a constant speed uis t=L/u,

so that mis given by

m(u)=t=L

uSw

|j·dA|.(6)

While this equation is adequate to study the behaviour of solid bodies enclosed in plane

surfaces (e.g. a parallelepiped), in the presence of curved surfaces it is in general more

convenient to write it in a different way, using the following theorem.

The absolute value of the ﬂux of a uniform vector ﬁeld over a surface, S, such that

each ﬁeld line intersects the surface at most once, is given by the product of the ﬁeld

intensity by the area of the projection, Spr , of the surface S on a plane orthogonal to

the ﬁeld.

The proof follows from the deﬁnition of the ﬂux, or Gauss’s theorem. Using this result,

the water mass can be written as

m(u)=L

u|j(u)|Spr (u). (7)

The time L/uclearly decreases monotonically with u. The projected wet surface

Spr depends on the shape of the body and on its orientation with respect to the rain and

the direction of motion—we cannot say anything in general about its dependence on u.We

note also that the projection of the wet surface exactly overlaps with the projection of the

whole body surface.

The absolute value of j(see equation (2)) increases monotonically with uif vx<0.

We know, when travelling in the rain, that if we speed up, the intensity of the rain seems to

increase. On the contrary, if vx>0, the apparent rain density has a minimum for u=vx.

The ratio |j|/(u)does not depend on the body: it is therefore connected with some general

features of the problem and it is worthwhile to study its dependence on u. Let us write it in a

more explicit form:

|j|

u=

ρ(vx−u)2+v2

y+v2

z

u=ρv

u2

−2vx

u+1=ρv

u2

−2v

ucos θ0+1.(8)

It is remarkable that this function, while monotonically decreasing for vx<0, always has

a minimum when vx>0, that is for

u=v2

vx

=v

cos θ0

.(9)

Aplotof|j|/(u)versus u/v for some values of θ0is shown in ﬁgure 2. We can see that for

θ0=30◦the function has a well pronounced minimum, while for 60◦the minimum is already

hard to distinguish.

This approach automatically solves the problem for all situations where the projected wet

surface does not depend on u. Of course, the simplest example one can consider is a sphere,

Whether or not to run in the rain 1325

30°

60°

110°

140°

Figure 2. Plot of the ratio |j|/(ρu)(arbitrary units) versus u/v for the following values of the angle

θ0between uand j0:30

◦and 60◦(tailwind), 110◦and 140◦(headwind).

for which the projected wet surface is always πR2,Rbeing the sphere’s radius. But this is not

the only case: as we have seen, when uchanges, jrotates in a plane perpendicular to the ζ

axis, so every body with a symmetry axis along this direction is in the same situation.

In conclusion, we can say that in all the cases where the projected wet surface does not

depend on u.

•An optimal speed exists, and its value is always given by equation (9), irrespective of the

body shape (cylinder, cone, ellipsoid ...), under the only condition that the wind comes

from behind. We note that the uvalue (9) is always greater than vx. Thus the optimal speed,

when it exists, is not always equal to vx, as found by previous authors.

•The optimal speed depends on v, and then on the drop size (since vobviously depends on

vz).

•An optimal speed exists irrespective of the intensity of the wind, provided that it has a tail

component.

Of course, in general Spr does depend on u, and its effect on m(u)maybeverydifferent

from one case to another. We can say that it is more relevant for bodies with a high degree of

asymmetry with respect to rotations around the ζaxis. We can be more precise only for some

particular, simple situations.

5. A parallelepiped and a plane surface

As said in the introduction, previous papers (with the exception of Ehrmann and Blachowicz

[2], who considered a vertical cylinder) have so far considered a parallelepiped with edges

parallel to Cartesian axes (we shall call it a ‘vertical’ parallelepiped) and a plane surface. Let

us now direct our attention to these two cases.

5.1. A vertical parallelepiped

The case of a vertical parallelepiped has been discussed by several authors, but we shall restate

the results brieﬂy here with our notation for the sake of convenience.

The rain ﬂux results:

(u)=ρ(Sx|vx−u|+Sy|vy|+Sz|vz|). (10)

1326 F Bocci

Here Sxdenotes the area of each face perpendicular to xaxis, and so on. As a consequence,

the mass of the intercepted water is:

m(u)=ρL

u(Sx|vx−u|+Sy|vy|+Sz|vz|). (11)

Depending on the value of vx, we must consider the following cases.

•vx0: headwind or no wind.

In this case |vx−u|=|vx|+u. Then:

m(u)=ρLSx+Sx|vx|+Sy|vy|+Sz|vz|

u.(12)

This function decreases monotonically, so the best strategy is to move as quickly as

possible.

•vx>0: tailwind.

At ‘low speed’ (u<v

x),|vx−u|=vx−u, so that

m(u)=ρL−Sx+Sx|vx|+Sy|vy|+Sz|vz|

u.(13)

This function, too, decreases monotonically, so it is never convenient to move at a speed

lower than vx. But what about a speed higher than vx?

At ‘high speed’ (uvx),|vx−u|=u−vx, then:

m(u)=ρLSx+−Sx|vx|+Sy|vy|+Sz|vz|

u(14)

and we must further differentiate between two subcases, depending on the sign of the

expression:

−Sx|vx|+Sy|vy|+Sz|vz|.(15)

When this expression is strictly positive, m(u)also decreases for uvx, so it is convenient,

once more, to move at a maximum speed. But when, on the contrary, it is negative, m(u)

increases for uvx, an optimal speed does exist and is always:

uopt =vx.(16)

We can observe that in this case the optimal speed depends only on the wind speed and

not on drop size. We note also that, when expression (15) vanishes, the mass of water

intercepted by the body is equal to the water found in the volume LSxand is independent

of u, provided only that u>v

x.

It is worthwhile to write down the condition that (15) be negative in two different ways:

Sx>Sx|vx|+Sy|vy

vx

,(17a)

vx>Sx|vx|+Sy|vy

Sx

.(17b)

Equation (17a) tells us that, for a given v, an optimal speed exists only for parallelepipeds

with a large Sx;(17b) tells that, for a given parallelepiped, an optimal speed exists only if vx

is large enough.

5.2. A generalisation?

For a parallelepiped, the rain ﬂux is the sum of three terms, each relative to a component of j:

=|jx|Sx+|jy|Sy+|jz|Sz.(18)

It is very tempting, at this point, to generalize this result to a body of any shape, by simply

deﬁning Sxto be the projected area of the body on a plane normal to xaxis, and so on, and

this seems to be the line followed by previous papers. However, even though very appealing,

Whether or not to run in the rain 1327

this idea is not correct, because the presence of the modulus in deﬁnition (5) does not permit

splitting of the sum of the component products.

We have already seen a simple example where equation (18) is not valid. In the case of a

sphere, according to this formula the ﬂux should be (|jx|+|jy|+|jz|)πR2, while it turns out

to be |j|πR2. Below we shall see some more examples.

5.3. A plane surface

In order to develop a deeper understanding of the physics of the problem, let us consider a

plane-oriented surface and Ato be the vector area associated with it. We need to consider two

more quantities: the angle, ϕ, between Aand j, whose initial value (the angle between Aand

j0) will be denoted by ϕ0, and the angle, ψ, between Aand u. Without any loss of generality,

we can always choose the orientation of Ain such a way that Axis positive. Equation (6) then

gives:

m(u)=L

u|j·A|= ρL

u|(v−u)·A|=ρL

v·A

u−Ax.(19)

From this relation we see the following.

•When v·A=0, that is when the surface is parallel to the rain, mdoes not depend on u,

so the speed does not matter at all. The surface simply collects all the water contained in

the volume LAx. In particular, when Ax=0, that is when the surface is in the πplane, it

remains dry.

•When v·A= 0 and Ax=0, the way to minimize mis to let ube as high as possible.

•Lastly, when v·Aand Axare both non-zero, the only relevant parameter is the angle

ϕ0, which determines the sign of the dot product v·A. Then we have that when

ϕ0is acute, it is always possible to choose a speed such that the surface does not

get wet at all. This optimal speed is clearly

uopt =v·A

Ax

=vcos ϕ0

cos ψ0

.(20)

When, on the contrary, ϕ0is obtuse, we see from equation (19) that m(u)decreases

monotonically, so the best strategy is again to move as quickly as possible (the surface

should move in the opposite direction in order to avoid getting wet).

It can be useful to plot expression (19), which is essentially the modulus of a vertically

shifted hyperbola branch, for the case of an acute and an obtuse value of ϕ0(see ﬁgure 3).

In conclusion, we can say that for a plane surface the choice of the best strategy depends

uniquely, for a given rain velocity, on the orientation of the surface with respect to the rain

(which determines the sign of the dot product v·A) and on the direction of motion (which

determines if Axvanishes or not).

Compared to the case of a vertical parallelepiped, for a plane surface the following

differences can be highlighted.

•When m(u)has a minimum, it is always zero, that is, the surface remains dry.

•An optimal speed can also exist with a headwind.

•There is no condition on vx; an optimal speed can also exist with a light wind.

•The optimal speed can be either lower or higher than vx.

•The value of uopt also depends on the modulus of the rain velocity, and then on the drop

size.

The condition that the angle ϕ0is acute has a simple physical meaning: given our

convention to orient Aso that Ax>0(A‘points ahead’), it simply means that the rain

wets the rear face of the surface. In this case, increasing u, the rotation of jwill ultimately

1328 F Bocci

120°

60°

Figure 3. Plot of m(arbitrary units) versus u/v for a plane surface in case of an acute (60◦)and

obtuse (120◦)angle ϕ0between vand A.

x

z

Rain

A

0 0

Rain

x

z

0

0

A

Figure 4. A plane surface parallel to the yaxis moves in the rain, in the absence of a lateral wind.

(Left) We have a tailwind and the angle ϕ0between the rain and Ais obtuse (the rain wets the front

face of the surface). Increasing u,jrotates clockwise and never becomes parallel to the surface,

so m(u)does not have a minimum. (Right) We have a headwind and the angle ϕ0is acute (the

rain wets the rear face). When jrotates, there is a particular (optimal) speed for which it becomes

parallel to the surface.

lead to wetting of the front face, and so there will be a particular value of ufor which the rain

will be parallel to the surface.

Now we can understand that the condition of having a tailwind is neither necessary nor

sufﬁcient to have a minimum in the function m(u). Figure 4explains this point for the cases

of vy=0 (no cross component of the wind) and Ay=0 (surface parallel to the yaxis).

From the discussion of this point—which concerns a very simple geometrical situation—

we can learn how complex the problem is and how we must pay attention to such aspects as

the orientation with respect to the direction of motion and of the rain.

6. From a plane surface to a solid body

Of course, a solid body is not a plane surface, it has several surfaces with different orientations,

so it is not surprising that results found in the two cases are different. For example, there is no

Whether or not to run in the rain 1329

A3

x

z

Rain

A1

0

0

Figure 5. A parallelepiped with two opposite faces parallel to the xz plane. By analogy with the

previously considered situation, we call ψthe angle between A1and the xaxis. We choose A1and

A3so that their xcomponent is positive.

way for a solid body to remain dry in the rain. However, the ‘transition’ from a plane surface

to a solid body, for physical reasons, must be smooth.

Let us consider a parallelepiped with a generic orientation. First of all we observe that

in a parallelepiped each pair of opposite faces behaves in the rain like one plane surface, so

a parallelepiped, for our discussion, is equivalent to three mutually perpendicular rectangles.

If we associate a vector area with each pair we get three mutually orthogonal vectors, that

we shall call A1,A2, and A3, and that we shall orient, as before, in such a way that their x

component is positive.

The equation that gives the water mass impinging on the parallelepiped is a simple

generalisation of equation (19), which reduces to (11) for a vertical parallelepiped:

m(u)=ρL

v·A1

u−A1x+

v·A2

u−A2x+

v·A3

u−A3x.(21)

This is the sum of three functions, each of which can have one of the two trends shown in

ﬁgure 2. The study of this function is a very complex problem, so we shall focus our attention

on a slightly simpler situation, that is an analogue of the ‘bi-dimensional’ case considered

in ﬁgure 3: a parallelepiped with two faces perpendicular to the yaxis (let us call A2the vector

area associated with this pair) and no cross component of the wind (vy=0)(see ﬁgure 5).

In this situation there is no ﬂux on the faces perpendicular to the yaxis. In order to shed

some light on the transition from a plane surface to a solid body, we may plot function (21)

for increasing values of the ratio A3/A1(see ﬁgure 6).

The plot has been obtained with the following parameters: θ0=125◦(headwind) and

ψ=50◦.

We can see how the thickness affects m(u): it gradually makes the minimum shallower, and

ultimately destroys it. The reason is clear; increasing the thickness has the effect of reducing

the asymmetry of the projected surface with respect to rotations in the πplane (in this case,

the xz plane). At the same time, it can be seen from this plot that there are cases where, even

for a solid body, a minimum exists with a headwind, and where the optimal speed is lower

than vx; some of the differences between a plane surface and a parallelepiped, described in

section 5.3, are just due to the orientation of the solid body.

1330 F Bocci

0

0.15

0.30

0.45

Figure 6. Plot of m(au) versus u/|vx|for a parallelepiped with ψ=50◦and rain falling at an

angle θ0=125◦(headwind), for various values of the ratio A3/A1: 0, 0.15, 0.30, 0.45.

j

x

y

z

Figure 7. A vertical cylinder. The wet surface is highlighted.

7. A cylinder

7.1. Vertical cylinder

Let us now consider a cylinder oriented vertically. In this case the wet surface Swis constituted

by the upper base and by one half of the lateral surface. To be more precise, it is one of the two

halves deﬁned by a plane passing through the cylinder axis and perpendicular to the horizontal

component of j, which we will denote jxy. The situation is shown in ﬁgure 7.

On the upper base, the only contribution to the ﬂux comes from jz. On the lateral surface,

on the other hand, only jxy contributes. With regard to this latter term, it is evident that the

associated projected surface Spr is a rectangle with base 2Rand height H,Rand Hbeing,

respectively, the radius and height of the cylinder. Then the rain ﬂux is:

(u)=ρπR2|vz|+2RH(vx−u)2+v2

y.(22)

Consequently, the water mass is:

m(u)=ρLR2π|vz|

u+2H

Rvx

u2

−2vx

u+1+v2

y.(23)

Whether or not to run in the rain 1331

Figure 8. Qualitative plots of m(arbitrary units) versus u/vxfor a vertical cylinder for various sets

of parameters. The upper left plot is identical to what can be obtained for a parallelepiped.

This expression is functionally different from that obtained for a parallelepiped, showing

one more case where equation (18) is not correct. Without performing a complete study of this

function, we can obtain some information from it.

First of all, we may note that for vx<0 this function is monotonically decreasing; a

minimum may exist only with a tailwind.

In case that vy=0, the cylinder behaves just like a parallelepiped with Ax=2RH and

Az=πR2. In this particular case, then, equation (18) may be applied. A vertical cylinder was

considered by Ehrmann and Blachowicz [2] who, however, did not take into account a cross

component of the wind and so missed the difference with a parallelepiped.

On the other hand, if vy= 0, we can plot the function (23) for various sets of parameters

in order to get some idea of its general behaviour (see ﬁgure 8).

Looking at these plots, we could conjecture the following.

•Increasing the ratios |vy|/vxor |vz|/vxhas the effect of making the minimum smoother and

of shifting the optimal speed towards higher values. If these ratios are too high, a minimum

may not exist at all.

•The same effects can be obtained by decreasing the ratio H/R. If this ratio is too low, a

minimum may not exist at all.

8. Conclusion

8.1. Results

Starting from very general assumptions, an equation that describes how to calculate the mass

of the water received by the body has been derived (we wrote it in two different ways:

equations (6) and (7)). We have solved or studied this equation for a plane surface and for

bodies with a simple shape. For a plane surface, we have found the following.

1332 F Bocci

•An optimal speed exists subject to the sole condition that the rain wets the rear face of the

surface, irrespective of the intensity of the wind and of the sign of vx.

•The optimal speed is uopt =v·A/Ax, so it depends on |v|, and then on the drop size.

•When moving at this optimal speed, the surface does not get wet.

•The optimal speed can have any value.

For solid bodies, we have solved the equation in all cases where the area of the wet surface

does not depend on u, that is when the body has a symmetry axis and this is perpendicular to

the plane identiﬁed by vand u. In these cases, an optimal speed exists, provided that the wind

has a component from behind; its value is uopt =v/ cos θ0, where θ0is the angle between v

and u, so in this case it also depends on the drop size.

The case of a vertical parallelepiped had been already studied and solved by previous

authors. We have extended this study by considering a slanted parallelepiped, and have found

that in some cases it is possible that an optimal speed exists even with a headwind, and that its

value is not always vx.

Lastly, we have considered a vertical cylinder, showing that its behaviour is different from

that of a vertical parallelepiped if there is a cross component of the wind velocity.

Our study of the behaviour of solid bodies in the rain has revealed a wide range of

situations, and general rules cannot be found. We can say that the presence of a tailwind seems

to be a favourable, but not always a necessary condition for the existence of an optimal speed.

In some cases, we have found that the value of uopt depends on the drop size, and in other

cases it does not.

8.2. Didactic considerations

While the problem seems to be too complex to be proposed as a didactic exercise, some

parts of it (e.g. the vertical parallelepiped, or the plane surface) could be suitable for didactic

purposes at the undergraduate level. First of all, the problem is familiar to all, and as such

stimulates students to apply themselves to ﬁnd the solution, and also to gaze around with a

‘physical mind’, looking for all the physical phenomena present in everyday life. Moreover, in

my opinion it is instructive to see how concepts (like ﬂux, current density, Gauss’s theorem)

turn out to be useful in other contexts. At the same time, it is beneﬁcial to think about the

fact that concepts are tools, and that they can and should be modiﬁed, if necessary, in order to

adapt them to our purposes, as we have with the ﬂux concept.

Acknowledgments

The author wishes to thank Paolo Violino and Germano Bonomi for a critical reading of the

manuscript and for useful comments, and an anonymous referee for considerable help and

many invaluable suggestions.

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