Unit 4: Section 2 - Gravitation And Electric Fields

Notes from The A2 revision guide

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Gravitational Fields

Any object with mass if affected by gravity, and will experience an attractive force when it is within the gravitational field of another object.

Gravitational lines of force are arrows that show the direction of the force that the masses will feel in a gravitational field.

If a small mass, m, enters the earth's gravitational field it will be attracted to it. Since the earth is much more massive than the mass, the mass will move towards the earth, but the earth will not move.

The further away from an object you get, the force experienced decreases.

F = - GMm / r²

M and m, are masses, G is the gravitational constant ( 6.67x10ֿ¹¹ ) and r is the distance between the two masses.

All objects have both a radial, and uniform field. Radial is from far away, uniform is from close up.

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Gravitational Fields Cont.

Gravitational field strength, g, is the force per unit mass. It is a vector quantity, always pointing towards the centre of the mass. It's units are Nkgֿ¹

g = F / m

The value of g at the earth's surface is 9.81msֿ² So an object in free fall at the earth's surface will accelerate at 9.81msֿ² due to gravity.

In a radial field, g is inversely proportional to r²

g = GM / r²

As it is an inverse square law, as r increases, g decreases


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Gravitational Fields Cont.

The gravitational potential, V, at a point is the gravitational potential energy that a unit mass at that point would have.

In a radial field, v = - GM / r

Gravitational potential is negative on the surface of the mass and increases with distance from the mass. Therefore at distance infinity, g = 0

g = - V / r

Two points at different distances from a mass will have different gravitational potentials, so there will be a gravitational potential difference. When you move an object you do work against gravity, The amount of energy required to move an object depends on the mass of the object, and the gravitational potential difference you move it through.

W = mV, Therefore W = m x GM / r = - GMm / r (Not to be confused with F = - GMm / r²)

A satellite is a smaller mass that orbits a larger mass. EG the moon around the earth.

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Gravitational Fields Cont.

The radius of an orbit effects an objects orbital period and speed.

An object in circular motion is kept in its path by a centripetal force. This force has to be equal to the gravitational force for the object to keep its orbit.

mv² / r = GMm / r²

v = GM / r

T = 2πr / v

T = √ 4π²r³ / GM

The greater the radius of a satellites orbit, the slower it will travel and the longer it will take to complete one orbit.

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Gravitational Fields Cont.

An orbiting satellite has both kinetic and potential energy, its total energy( KE + PE ) is always constant.

A geosynchronous satellite orbits the earth directly above the equator and stays above the same point all the time.

It travels at the same angular speed as the earth ( See circular motion notes ). The time taken for one complete orbit is 24 hours.

These types of satellites are useful for TV and telephone signals as since the satellite is always in the same positive, receivers and transmitters don't need to be constantly altered to keep receiving a signal.

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Electric Fields

Like charges repel, opposite charges attract.

When a charged object is placed within an electric field, it experiences a force.

Coulomb's law: F = (1 / 4πε) x ( Q1Q2 / r² )

ε is the permittivity of free space and equals 8.85x10ֿ¹² Fmֿ¹( Farad per metre )

The force on Q1 is always equal and opposite to the force on Q2. The direction depends on the changes.

Coulomb's law is an inverse square law. As r increases, F decreases.


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Electric Fields Cont.

The electric field strength, E, is the force per unit positive charge.

E = F / q

The units of E are Nm ֿ¹ and is a vector pointing in the direction that a positive charge would move. Field strength depends on the charges location within the field. Like gravitational fields, electric fields have a radial field.

In a radial field, E = (1 / 4πε) x ( Q / r² )

In a uniform field, E is inversely proportional to d. It also has the same value anywhere in the field between two parallel plates.

E = V / d


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Electric Fields Cont.

All points in an electric field have an electric potential, V. This is the electric potential energy that a unit positive charge ( +1C ) would have at that point.

In a radial field this is given by: E = (1 / 4πε) x ( Q / r )

For a repulsive force: V is initially positive and tends to zero as r increases towards infinity.

For a negative force: V is initially negative and tends to zero as r increases towards infinity.

Just like with gravitational fields, if two points in an electric field have different potentials then there is an electric potential difference between them. The amount of energy needed to move a charge in an electric field depends on the size of the charge and the potential difference it is moved across.

W = QV

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Electric Fields Cont.

There are four key similarities between electric and gravitational fields and three differences

Similarities:

1) g is force per unit mass, E is force per unit positive charge

2) The force in both is an inverse square law. F α 1 / r²

3) Field lines for gravitational field are the same as for a negative charge

4) Both potential energy's are infinity at point zero.

Differences:

1) Gravity always attracts, it never repels.

2) Objects can be shielded from electric fields

3) The size of an electric force depends on the medium between the charges ( EG. air )

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